Physics-Informed Neural Representation (PINN)

• Physics-Informed Neural Representation (PINN) is a neural network approach that embeds physical laws via differential equations directly into the training loss.
• It employs fully connected MLP architectures and automatic differentiation to accurately compute derivatives required for enforcing PDE constraints.
• PINNs enable forward simulation and inverse modeling for complex systems while addressing challenges in optimization, scalability, and loss-balancing.

Physics-Informed Neural Representation (PINN) refers to a class of neural architectures and training methodologies in which the governing equations of physics—typically in the form of partial differential equations (PDEs) or ordinary differential equations (ODEs)—are directly incorporated into the loss function used to train neural networks. This enables strong data efficiency, flexible solution representation, and natural blending of physical knowledge with measurement data, thereby facilitating both forward simulation and inverse inference for complex physical systems.

1. Mathematical Principles and General Framework

The core mathematical principle behind PINNs is the embedding of physical constraints—encoded by differential equations and initial/boundary conditions—into the optimization objective for a neural network surrogate. For a generic time-dependent PDE posed over domain $\Omega \times [0,T]$ with prescribed Dirichlet, Neumann, and initial conditions, and (optionally) scattered data observations, the canonical PINN loss function takes the form (Coulaud et al., 2024):

$$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$

The composite loss minimized during training is a sum of mean-square-error terms:

$$L = L_{\mathrm{PDE}} + L_{\mathrm{I}} + L_{\mathrm{D}} + L_{\mathrm{N}} + L_{\mathrm{data}}$$

where, for example,

$$L_{\mathrm{PDE}} = \frac{1}{N_\mathrm{PDE}} \sum_{i=1}^{N_\mathrm{PDE}} \left| \frac{\partial u}{\partial t} + \mathcal{N}(u,\nabla u) \right|^2,$$

and similarly for $L_{\mathrm{I}}$ (initial), $L_{\mathrm{D}}$ (Dirichlet), $L_{\mathrm{N}}$ (Neumann), $L_{\mathrm{data}}$ (data observations).

Spatial and temporal derivatives are either computed via automatic differentiation, as is standard, or (alternatively) using finite-difference stencils (Lim et al., 25 Feb 2026).

2. Neural Architectures and Solution Representation

The most prevalent PINN architecture is the fully connected multi-layer perceptron (MLP), which maps problem coordinates (and optionally problem parameters) to solution fields (Coulaud et al., 2024, Kag et al., 2023). For time-dependent PDEs, the input layer accepts variables such as $(x, t)$, and the output layer produces quantities of interest—velocity, pressure, temperature, or other field values. For multiphysics or multi-output settings (e.g., Navier–Stokes with energy or solid mechanics), the network can simultaneously output multiple physical fields. Activation functions are generally smooth and differentiable, with $\tanh$, sinusoidal, GELU, and Swish commonly used for their derivative properties. Advanced variants use architectures with gated or attention mechanisms (LDA-PINN, xLSTM-PINN), residual connections, or integrated physics-motivated output layers (e.g., Lehmann representation for quantum impurity problems) (Niu et al., 19 Jan 2026, Tao et al., 16 Nov 2025, Kakizawa et al., 2024).

For certain tasks (e.g., parametric surrogates), a parameter $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$0 is included as an additional input to model families of PDE solutions as $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$1 (Coulaud et al., 2024).

3. Parametric Surrogates and Multiphysics Extensions

PINNs naturally support the construction of parametric surrogate models: by introducing physical or geometric parameters as input, the network learns a solution operator over a prescribed range (e.g., viscosity or thermal conductivity in flow problems, or elastic moduli in solid mechanics) (Coulaud et al., 2024, Kag et al., 2023). Training involves sampling $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$2 tuples and minimizing loss over the joint domain.

Multiphysics coupling can be achieved via network composition or domain decomposition strategies. For example, conjugate heat transfer is implemented by training two PINNs: a fluid PINN enforcing Navier–Stokes and energy equations, and a solid PINN enforcing the Laplace heat equation. Coupling is realized by enforcing interface continuity of temperature and heat flux via additional loss terms (Coulaud et al., 2024).

4. Variants: Optimization, Architecture Search, and Spectral Mitigation

Training PINNs is nontrivial due to highly nonconvex loss landscapes, multiple competing constraints, and ill-conditioning. Standard optimizers (Adam, SGD) often stagnate; convergence and accuracy are enhanced by quasi-Newton solvers (L-BFGS, custom BFGS, self-scaled Broyden), but these preclude minibatching and may not scale to large parameter sets or high-dimensional problems (Coulaud et al., 2024, Arnaud et al., 22 Apr 2026).

Neural architecture search (NAS) methods—such as NAS-PINN and Auto-PINN—systematically optimize depth, width, activation type, and more, using bilevel optimization or sequential search to minimize PINN loss on held-out validation sets. Findings consistently show that increased depth does not guarantee improved error and that each PDE exhibits an optimal architecture pattern, often requiring moderate depth and nonuniform width (Wang et al., 2023, Wang et al., 2022).

Innovations such as dynamic attention mechanisms (LDA-PINN), memory-gated RNN architectures (xLSTM-PINN), and gradient conflict resolution (PCGrad, ACR-PINN) have been shown to improve convergence rates and reduce $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$3 errors by 1–2 orders of magnitude on benchmark PDEs (Tao et al., 16 Nov 2025, Niu et al., 19 Jan 2026).

Spectral bias—the poor representation and learning rate of high-frequency components—can be mitigated both by representation-level remodeling (xLSTM, frequency curriculum) and by transfer-learning strategies that gradually increase solution complexity (from low- to high-frequency regimes) (Tao et al., 16 Nov 2025, Mustajab et al., 2024).

5. Data Assimilation and Inverse Modeling

PINNs facilitate data assimilation and inverse parameter inference by promoting unknown physical parameters, coefficients, or even field variables (e.g., turbulent viscosity, elastic moduli) to trainable variables. A joint optimization is performed over both network parameters $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$4 and the unknown physical quantity $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$5, with gradients computed via automatic differentiation (Coulaud et al., 2024, Kag et al., 2023). In turbulence modeling (e.g., backward-facing step flows), the network simultaneously infers mean velocity, pressure, and turbulence closure coefficients. In solid mechanics, PINNs estimate material constants from sparse observations with $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$6 error (Kag et al., 2023).

Hard-constrained approaches using variants of trust-region sequential quadratic programming (trSQP-PINN) replace the penalty-based losses by directly imposing the PDE and boundary residuals as constraints, achieving 1–3 orders of magnitude better accuracy in challenging regimes (Cheng et al., 2024).

6. Robustness, Accuracy, and Scalability

Robustness studies consistently show that PINN prediction accuracy is sensitive to the choice of optimizer, network architecture, sampling scheme, and loss weighting (Coulaud et al., 2024). For model problems in aerodynamics and solid mechanics, field variable errors are typically well below $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$7, with interface and residual errors in the $$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$8–$$\begin{align*} \text{PDE:} & \quad \frac{\partial u}{\partial t} + \mathcal{N}(u, \nabla u) = 0, \ \text{Dirichlet:} & \quad u(x^D, t) = g^D(x^D, t), \ \text{Neumann:} & \quad \frac{\partial u}{\partial n}(x^N, t) = g^N(x^N, t), \ \text{Initial:} & \quad u(x, 0) = g^I(x), \ \text{Data:} & \quad u(x_i, t_i) = u_i^\star, \quad i=1 \ldots N_\mathrm{data}. \end{align*}$$9 range. Data efficiency is observed: increasing data coverage offers only marginal gains after a threshold, but inadequate coverage (e.g., absence of recirculation zone data) leads to substantial errors (Coulaud et al., 2024, Kag et al., 2023).

Computational scaling, particularly for large 3D or turbulent-flow problems, is limited by the need for large full-batch AD sweeps and second-order optimization. Current remedies include adaptive collocation (sampling more where residuals are highest), mask-based loss weighting at sharp boundaries, and decomposition into multiple smaller sub-networks (Arnaud et al., 22 Apr 2026).

7. Limitations, Open Challenges, and Future Directions

Despite demonstrated flexibility and accuracy, current PINN methodology faces several open challenges (Coulaud et al., 2024, Ganga et al., 2024):

• Loss function ill-conditioning and the need for dynamic or adaptive balancing of loss weights.
• Sensitivity to optimizer and hyperparameter choices; automated architecture and optimizer selection (NAS, Auto-PINN, PCGrad) is promising but not yet universal.
• Computational bottlenecks due to repeated AD and lack of efficient minibatching for quasi-Newton methods.
• Scalability limitations on large, high-frequency, or high-Reynolds-number problems.
• Enforcement of complex or mixed boundary conditions in irregular domains, although hybrid approaches (PINN-FEM) have achieved exact Dirichlet satisfaction (Sobh et al., 14 Jan 2025).

Proposed directions include the development of adaptive loss-weighting schemes, second-order optimizers tailored for PINNs, theory-informed parametric neural surrogates (e.g., DeepONet, FNO), incorporation of domain decomposition (XPINN/APINN), embedding analytic/spectral structure in network designs, and transfer learning or frequency curricula for multiscale problems (Tao et al., 16 Nov 2025, Mustajab et al., 2024).

PINNs provide a unifying meshless, physics-constrained neural framework for addressing forward, inverse, and optimal control problems in computational physics, with ongoing progress toward improving stability, scalability, and theoretical understanding (Coulaud et al., 2024, Kag et al., 2023, Ganga et al., 2024).