Quasi-Random PINNs: Enhanced Sampling for PDEs

• Quasi-Random PINNs are physics-informed neural networks that replace random collocation with low-discrepancy sampling (e.g., Halton, Sobol’) to better approximate continuous residual objectives.
• They retain the standard PINN architecture and loss formulation while refining the sampling stage to reduce quadrature error, especially in high-dimensional problems.
• Empirical results demonstrate that QRPINNs can yield up to a 32.8% error reduction in high-dimensional settings compared to traditional random sampling methods.

Searching arXiv for the primary QRPINN paper and closely related PINN collocation literature. Physics-informed neural networks (PINNs) approximate solutions of differential equations by minimizing losses that penalize violations of governing equations, initial conditions, and boundary conditions at sampled points in the domain. Quasi-Random Physics-informed Neural Networks (QRPINNs) modify this standard PINN workflow at the sampling stage: instead of drawing collocation points directly from the domain by ordinary random sampling, they use low-discrepancy sequences, specifically quasi-Monte Carlo constructions such as Halton and Sobol’, or random minibatches drawn from pools generated by those sequences (Yu et al., 10 Jul 2025). The core motivation is that the PINN loss is an empirical quadrature of a continuous residual-based objective, so the distribution of collocation points directly affects how accurately the discrete loss approximates its integral counterpart (Yu et al., 10 Jul 2025). In the broader PINN literature, the baseline formulation is explicitly built around enforcing physics on “random points in the space-time domain,” which makes QRPINNs a sampling-level refinement rather than a change to the PINN architecture or residual definition (Shukla et al., 2022).

1. Definition and conceptual position

QRPINNs are a variant of PINNs in which “the usual random sampling of collocation/training points is replaced by sampling from low-discrepancy, quasi-random point sets” (Yu et al., 10 Jul 2025). In ordinary PINNs, the approximation $\boldsymbol{u}^\theta$ is trained by minimizing residual, initial-condition, and boundary-condition losses evaluated at sampled points from the corresponding domain subsets. QRPINNs “keep the PINN architecture and objective essentially unchanged; the key modification is the generation of training/collocation points” (Yu et al., 10 Jul 2025).

This places QRPINNs within the standard mesh-free PINN paradigm rather than in alternative physics-informed learning frameworks. The review literature describes classical PINNs as methods that “can solve a partial differential equation (PDE) by directly incorporating the PDE into the loss function of the neural network (NN) and employing automatic differentiation to represent all the differential operators,” with the physics enforced by penalizing residuals at random collocation points (Shukla et al., 2022). QRPINNs preserve this formulation and intervene only in the construction of the sampled point sets. This suggests that QRPINNs are best understood as a collocation-design modification of PINNs, analogous in role to changing the quadrature rule used to discretize a continuous residual norm.

A frequent source of confusion is the abbreviation “QR.” In QRPINNs, it refers to “quasi-random,” whereas in the adaptive collocation method based on QR Discrete Empirical Interpolation Method (QR-DEIM), “QR” refers to QR factorization with column pivoting rather than low-discrepancy sampling (Celaya et al., 13 Jan 2025). The two lines of work address the same broad question—how to place collocation points in PINNs—but they are not the same method.

2. Mathematical formulation

The standard time-dependent PDE form used in the QRPINN paper is

$$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$

with neural approximation $\boldsymbol{u}^\theta$ and composite loss

$$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$

where

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$

In standard PINNs, the point sets $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$ are typically sampled uniformly at random from the corresponding domain pieces (Yu et al., 10 Jul 2025).

The QRPINN paper interprets these finite sums as empirical approximations to an “infinite-sample” objective

$$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$

with

$$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$

For the residual term, the paper writes the empirical average as converging to the corresponding space-time integral:

$$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$

This loss-as-quadrature perspective is the formal entry point for QRPINNs (Yu et al., 10 Jul 2025).

The broader PINN review presents the same structure in a generic PDE form on $\Omega$, with residual points $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$0 and boundary points $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$1 and total loss

$$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$2

where the paper states explicitly: “To evaluate the residual of the PDE and the boundary/initial conditions, we introduce a set of random points inside the domain ($$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$3) and another random set of points on the boundary ($$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$4)” (Shukla et al., 2022). In a QRPINN, the loss decomposition is not changed; only the generation of $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$5 and $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$6 is changed.

3. Quasi-random sampling and theoretical rationale

The specific low-discrepancy sequences implemented in the QRPINN paper are Halton and Sobol’ (Yu et al., 10 Jul 2025). For Halton, with prime bases $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$7, the points are

$$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$8

with

$$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$9

For Sobol’, the paper gives

$\boldsymbol{u}^\theta$0

where $\boldsymbol{u}^\theta$1 is the dyadic expansion and

$\boldsymbol{u}^\theta$2

with $\boldsymbol{u}^\theta$3 defined recursively from primitive polynomials over $\boldsymbol{u}^\theta$4 (Yu et al., 10 Jul 2025).

The paper’s theoretical motivation begins from the Monte Carlo approximation

$\boldsymbol{u}^\theta$5

for which the PINN paper emphasizes the usual Monte Carlo convergence rate $\boldsymbol{u}^\theta$6 (Yu et al., 10 Jul 2025). It also gives

$\boldsymbol{u}^\theta$7

For quasi-Monte Carlo, the paper states a better rate, generally

$\boldsymbol{u}^\theta$8

and in weighted settings

$\boldsymbol{u}^\theta$9

For Halton and Sobol’, it mentions the classical form

$$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$0

The paper reproduces a discrepancy-based inequality. In one dimension,

$$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$1

and for $$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$2,

$$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$3

where $$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$4 is the star discrepancy. The star discrepancy for a point set $$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$5 is defined by

$$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$6

where $$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$7, $$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$8, and

$$\mathcal{L}(\theta)=\mathcal{L}_{i c}(\theta)+\mathcal{L}_{b c}(\theta)+\mathcal{L}_r(\theta),$$9

The intended conclusion is that low-discrepancy point sets reduce the mismatch between empirical point coverage and uniform volume, thereby improving the approximation of the continuous PINN objective by the discrete loss (Yu et al., 10 Jul 2025).

This argument is tied to a theorem in the QRPINN paper:

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$0

such that for $$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$1,

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$2

The paper uses this to link PINN approximation quality to quadrature error: once network approximation error is controlled, the gap between the empirical minimizer and the ideal integral-loss minimizer is of the same order as the loss-approximation error (Yu et al., 10 Jul 2025). This suggests that improving the sampling rule may improve the overall solution by improving the quadrature stage.

4. Randomized quasi-Monte Carlo training

The QRPINN paper does not advocate training on one fixed deterministic low-discrepancy set at every epoch. Instead, it introduces Random Quasi Monte Carlo (RQMC): a deterministic low-discrepancy pool is generated first, and minibatches are then sampled randomly from that pool during training (Yu et al., 10 Jul 2025). The authors summarize the method by stating that QRPINNs “randomly sample points from low-discrepancy sequences rather than the input domain” (Yu et al., 10 Jul 2025).

This design reflects a specifically machine-learning-oriented compromise. Fully deterministic QMC preserves discrepancy properties, but minibatch stochasticity is often useful in neural optimization. The paper therefore frames QRPINNs as standard PINNs trained with an RQMC sampling scheme rather than as fully deterministic quadrature solvers.

The associated theorem is stated as:

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$3

if $$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$4 is a subset sampled from $$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$5, and if

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$6

for some $$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$7, and

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$8

then

$$\begin{aligned} & \mathcal{L}_r(\theta)=\frac{1}{N_r} \sum_{i=1}^{N_r}\varepsilon_r(\theta,t_r^i, \boldsymbol{x}_r^i), \quad \varepsilon_r(\theta,t,\boldsymbol{x})=\left|\partial_{t}\boldsymbol{u}^\theta\left(t,\boldsymbol{x}\right)+\mathcal{N}\left[\boldsymbol{u}^\theta\right]\left(t,\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{i c}(\theta)=\frac{1}{N_{i c}} \sum_{i=1}^{N_{i c}}\varepsilon_{ic}\left(\boldsymbol{x}_{i c}^i\right), \quad \varepsilon_{ic}\left(\boldsymbol{x}\right)=\left|\boldsymbol{u}^\theta\left(0, \boldsymbol{x}\right)-\boldsymbol{g}\left(\boldsymbol{x}\right)\right|^2,\ & \mathcal{L}_{b c}(\theta)=\frac{1}{N_{b c}} \sum_{i=1}^{N_{b c}}\varepsilon_{bc}\left(t_{b c}^i, \boldsymbol{x}_{b c}^i\right), \quad \varepsilon_{bc}\left(t, \boldsymbol{x}\right)=\left|\mathcal{B}\left[\boldsymbol{u}^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2. \end{aligned}$$9

for constants $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$0, $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$1, and dimensionality $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$2 (Yu et al., 10 Jul 2025). The paper notes notation inconsistencies in the theorem statement, but the intended message is that random minibatching from a large low-discrepancy pool can preserve favorable asymptotic behavior.

The paper further adds a combinatorial coverage argument:

$$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$3

and states that this is “almost 1” when $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$4 and $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$5, which it uses to argue that $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$6 is practical and that the error convergence of RQMC is almost the same as QMC (Yu et al., 10 Jul 2025).

A notable practical hyperparameter is the sampling-pool scale $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$7. In the main 100D experiments the QRPINN paper sets $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$8 and trains for 30 epochs (Yu et al., 10 Jul 2025). It also reports that performance depends on $$\boldsymbol{x}_{ic}, \boldsymbol{x}_{bc}, \boldsymbol{x}_r$$9, with Halton more sensitive than Sobol’, and emphasizes that using a larger quasi-random pool with stochastic minibatching can outperform fully deterministic training (Yu et al., 10 Jul 2025).

5. Empirical performance and dimensional regime

The QRPINN paper studies steady Poisson, steady Allen-Cahn, steady Sine-Gordon, and time-dependent high-dimensional PDEs coupled with the Stochastic Taylor Derivative Estimator (STDE), including semi-linear heat, time-dependent Allen-Cahn, and time-dependent Sine-Gordon (Yu et al., 10 Jul 2025). The steady experiments consider $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$0, while the STDE experiments extend to dimensions $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$1 (Yu et al., 10 Jul 2025). Baselines are Vanilla PINNs with uniform random sampling, RAD, ACLE, Halton-based QRPINN, and Sobol’-based QRPINN (Yu et al., 10 Jul 2025).

The empirical pattern is dimension-dependent. In lower dimensions, QRPINNs are competitive but not universally dominant. For example, in 3D Allen-Cahn the best result reported is Vanilla at $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$2, with Halton very close at $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$3; in 10D Allen-Cahn, Vanilla is again best at $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$4, while the other methods are described as very close (Yu et al., 10 Jul 2025). In 3D Sine-Gordon, Sobol’ is best at $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$5, whereas in 10D Sine-Gordon, Vanilla is best at $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$6, with Halton and Sobol’ only slightly worse (Yu et al., 10 Jul 2025).

The paper identifies stronger evidence in 100D. Compared with the best result among Vanilla and RAD, QRPINNs improve by 22.6% for Allen-Cahn and 32.8% for Sine-Gordon (Yu et al., 10 Jul 2025). The supporting values given in the appendix are:

Problem	Setting	Comparison
Allen-Cahn	100D, 1000 points	Vanilla $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$7, RAD $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$8, Halton $$\begin{aligned} \mathcal{L}_{int}(\theta):= & \int_t\int_{\Omega} \left|\partial_{t}u^\theta\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t \ & +\int_\Omega \left|u^\theta\left(0, \boldsymbol{x}\right)-g\left(\boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x} \ & + \int_t\int_{\partial\Omega}\left|\mathcal{B}\left[u^\theta\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t, \end{aligned}$$9
Sine-Gordon	100D, 1000 points	Vanilla $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$0, RAD $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$1, Halton $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$2

The STDE results push the method into much higher dimensions and are the strongest support for the claim that QRPINNs become more attractive as $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$3 grows. The paper reports the following examples (Yu et al., 10 Jul 2025):

Problem	Dimension	Reported values
Allen-Cahn	$$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$4	Vanilla $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$5, Halton $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$6, Sobol $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$7
Allen-Cahn	$$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$8	Vanilla $$\theta^*_\infty := \arg \min_\theta \mathcal{L}_{int}(\theta).$$9, Sobol $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$0
Heat	$$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$1	Vanilla $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$2, Sobol $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$3
Sine-Gordon	$$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$4	Vanilla $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$5, Sobol $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$6

The paper also records failure cases. For Sine-Gordon at $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$7, Vanilla is best and QRPINNs do not improve it; for Heat at $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$8 and $$\frac{1}{N_r} \sum_{i=1}^{N_r}\left|\partial_{t}u\left(t_r^i, \boldsymbol{x}_r^i\right)+\mathcal{N}\left[u\right]\left(t_r^i, \boldsymbol{x}_r^i\right)\right|^2 \rightarrow \int_t\int_{\Omega} \left|\partial_{t}u\left(t, \boldsymbol{x}\right)+\mathcal{N}\left[u\right]\left(t, \boldsymbol{x}\right)\right|^2\mathrm{d}\boldsymbol{x}\mathrm{d}t.$$9, neither Halton nor Sobol beats Vanilla (Yu et al., 10 Jul 2025). This matters because it limits the scope of the empirical claim: QRPINNs are not presented as uniformly superior, but as increasingly favorable in high-dimensional regimes.

The evaluation metric throughout is relative error, reported as mean $\Omega$0 standard deviation over repeated runs, with experiments marked as failures when the relative error exceeds $\Omega$1 (Yu et al., 10 Jul 2025).

6. Relationship to adaptive collocation methods

The literature surrounding QRPINNs repeatedly distinguishes static space-filling designs from adaptive refinement. The review article on scalable PINNs describes the classical baseline entirely in terms of random collocation points and discusses extensions such as self-adaptive weighting, gradient-enhanced PINNs, and domain decomposition, but “there is no explicit discussion of quasi-random sampling, low-discrepancy sequences, Latin hypercube sampling, stratified sampling, importance sampling, residual-based adaptive refinement, or quadrature-based collocation” (Shukla et al., 2022). This makes the review a baseline reference for standard random-point PINNs rather than a source on QRPINNs themselves.

The adaptive QR-DEIM paper is more directly relevant to sampling strategy. It distinguishes fixed methods—uniform random sampling, equispaced grids, Latin hypercube sampling, Sobol sequences, and Hammersley sequences—from adaptive ones, and argues that fixed methods are “agnostic to the evolving PDE residual” and can under-resolve regions with high gradients, phase boundaries, shocks, localized complex dynamics, and later times in time-dependent problems (Celaya et al., 13 Jan 2025). Its proposed method is not quasi-random; instead, it builds a residual snapshot matrix over a training window, compresses it with SVD, and selects informative points by QR with column pivoting (Celaya et al., 13 Jan 2025).

The QR-DEIM paper’s core workflow is based on a training set $\Omega$2 and a disjoint snapshot set $\Omega$3, with residual snapshots collected over $\Omega$4 iterations,

$\Omega$5

followed by

$\Omega$6

and QR with column pivoting on the reduced basis transpose,

$\Omega$7

The selected snapshot-point indices are then added to the training set (Celaya et al., 13 Jan 2025). The paper argues that this improves accuracy over uniform random sampling and representative residual-based adaptive methods on wave, Allen-Cahn, and Burgers’ equations, and reports that the method tends to place more points near final time, near phase boundaries in Allen-Cahn, and around the shock in Burgers while still maintaining some global coverage (Celaya et al., 13 Jan 2025).

For QRPINNs, the main significance of this literature is comparative rather than definitional. The QRPINN paper itself proposes combining low-discrepancy pools with RAD-style adaptive sampling and reports that, for 100D Poisson, replacing the RAD candidate pool by low-discrepancy points often helps. It gives, for example, $\Omega$8: Vanilla+RAD $\Omega$9, Halton+RAD $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$00, Sobol+RAD $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$01; and for $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$02: Vanilla+RAD $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$03, Halton+RAD $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$04, Sobol+RAD $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$05 (Yu et al., 10 Jul 2025). It also states that low-discrepancy pools help RAD more consistently than ACLE, while for ACLE they can be invalid and decrease accuracy (Yu et al., 10 Jul 2025).

This suggests a division of labor. QRPINNs address the quality of global initial coverage through low discrepancy, whereas adaptive methods address evolving localized difficulty during training. The adaptive collocation paper makes this logic explicit by arguing that “static space-filling alone is not enough for difficult PDEs” and that collocation points should concentrate near shocks, interfaces, sharp fronts, localized layers, and late-time regions (Celaya et al., 13 Jan 2025). A plausible implication is that quasi-random collocation and residual-driven adaptivity are complementary rather than mutually exclusive.

7. Computational characteristics, limitations, and scope

The QRPINN paper argues that generating low-discrepancy pools is inexpensive relative to PINN training. It reports roughly linear runtime scaling in the number of points and linear memory scaling in both $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$06 and $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$07, with Sobol’ more sensitive to dimensionality in runtime than Halton at high dimension (Yu et al., 10 Jul 2025). For example, at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$08, Halton generation time is about $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$09 ms at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$10, $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$11 ms at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$12, and $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$13 ms at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$14; Sobol’ is about $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$15 ms at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$16, $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$17 ms at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$18, and $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$19 ms at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$20 (Yu et al., 10 Jul 2025). The overhead is therefore described as small but not zero.

The method’s limitations are also stated directly. First, only two classical low-discrepancy sequences, Halton and Sobol’, are implemented; modified Halton, Niederreiter, FNN, component-by-component constructions, random lattice rules, and weighted QMC are suggested as future possibilities but are not tested (Yu et al., 10 Jul 2025). Second, generating very large quasi-random pools becomes increasingly costly in extremely high dimensions, so the paper does not claim efficiency indefinitely as $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$21 (Yu et al., 10 Jul 2025). Third, QRPINNs are “not universally dominant”: in lower-dimensional problems, especially when locality is strong, adaptive samplers such as RAD or ACLE may outperform them (Yu et al., 10 Jul 2025). Fourth, combining low-discrepancy pools with all adaptive samplers is not always beneficial; ACLE is given as a negative example (Yu et al., 10 Jul 2025). Fifth, the paper notes that some theorem statements and bounds are “notationally rough,” so the theoretical claim is stronger at the level of intended mechanism than at the level of formal polish (Yu et al., 10 Jul 2025).

The broader PINN literature reinforces that sampling is only one component of a larger systems problem. The review article identifies “the large computational cost associated with the training of the neural networks, especially for forward multi-scale problems” as a major concern, and emphasizes multiscale difficulty, stiffness, loss-weight choice, and scalability through domain decomposition such as cPINN and XPINN (Shukla et al., 2022). This suggests that QRPINNs target one recognized bottleneck—collocation quality—without claiming to solve all PINN failure modes.

A concrete non-QRPINN example helps delimit the expected regime of benefit. In the Teukolsky-equation PINN for Kerr black-hole quasi-normal modes, the collocation scheme is a fixed uniform grid with $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$22 over two one-dimensional domains, and the authors explicitly state that “a uniform grid is not a requirement” (Luna et al., 2022). Because that problem involves one-dimensional radial and angular variables, the paper notes that uniform grids are already strong baselines; this suggests that the largest gains from quasi-random sampling may appear not in simple 1D settings, but in higher-dimensional or more stochastic PINN regimes (Luna et al., 2022). The QRPINN paper’s own results, which are strongest at $$\begin{aligned} & \partial_{t}\boldsymbol{u}+\mathcal{N}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \Omega, \ & \boldsymbol{u}(0, \boldsymbol{x})=\boldsymbol{g}(\boldsymbol{x}), \quad \boldsymbol{x} \in \Omega, \ & \mathcal{B}[\boldsymbol{u}]=0, \quad t \in[0, T],\ \boldsymbol{x} \in \partial \Omega, \end{aligned}$$23 and beyond, are consistent with that pattern (Yu et al., 10 Jul 2025).

In the current literature, QRPINNs are therefore best characterized as a minimal modification of PINNs motivated by quasi-Monte Carlo integration: retain the standard residual-based objective and neural architecture, but replace direct random domain sampling with random minibatching from low-discrepancy pools such as Halton or Sobol’ (Yu et al., 10 Jul 2025). The published evidence indicates that this modification is especially relevant for high-dimensional PDEs, where the mismatch between empirical collocation coverage and the continuous residual objective becomes increasingly consequential (Yu et al., 10 Jul 2025, Shukla et al., 2022).