Discontinuity-aware PINN (DPINN)

• DPINN is a class of physics-informed neural network formulations designed to handle discontinuities and sharp gradients in PDE solutions.
• It employs techniques like domain decomposition, adaptive Fourier features, and residual reweighting to mitigate spectral bias and optimization challenges.
• DPINNs extend the solvable regime for advection-dominated and interface problems, achieving significant accuracy improvements over standard PINNs.

Discontinuity-aware Physics-Informed Neural Network (DPINN) denotes a class of physics-informed neural-network methods designed for partial differential equations whose solutions contain shocks, contacts, interfaces, boundary layers, or other sharp gradients for which standard smooth-network PINNs are known to perform poorly. In the recent literature, the term has two related but distinct usages. In (Lei et al., 28 Nov 2025) and (Lei et al., 11 Jul 2025), DPINN explicitly means Discontinuity-aware Physics-Informed Neural Network. In (Rout et al., 2021), by contrast, DPINN means Distributed Physics Informed Neural Network, yet its purpose is likewise to address advection-dominated problems with strong boundary layers and near-discontinuities by decomposing the domain into subdomains. Taken together, these works identify a common design objective: replace globally smooth, strong-form residual minimization by architectures, losses, and interface treatments that localize sharp features, regularize singular residuals, and encode the correct weak or interfacial physics (Rout et al., 2021).

1. Terminology and conceptual scope

A discontinuity-aware PINN is motivated by a recurrent failure mode of classical PINNs: the network, residual, and optimizer are all biased toward smooth solutions, whereas many conservation laws and interface problems admit only weak solutions with jumps or very steep transition layers. Several papers in the supplied corpus describe this failure in different domains. For advection-dominated advection–diffusion, a global PINN struggles to represent a sharp boundary layer and becomes ill-conditioned as the diffusion coefficient decreases (Rout et al., 2021). For Burgers and Euler equations with shocks, strong-form PINNs are sensitive to discontinuities and can produce smeared or unstable shock profiles unless the formulation is altered (Neelan et al., 27 Jun 2025). For phase-field multiphase flow, standard PINNs suffer from spectral bias, gradient-direction conflict, and long-time error accumulation when interfaces are thin and highly nonlinear (Lei et al., 28 Nov 2025). For Maxwell problems in discontinuous media, smooth networks underresolve interface-induced high frequencies unless the interface geometry is explicitly encoded (Nohra et al., 2024).

The literature therefore treats “discontinuity-aware” not as a single algorithm but as a design principle. A DPINN modifies at least one of the following: the function representation, the loss functional, the domain decomposition, the numerical flux or jump treatment, the viscosity or regularization mechanism, or the training schedule. This suggests that DPINN is best understood as a family of PINN formulations for weak-solution regimes rather than a single canonical architecture.

A terminological ambiguity is important. The thesis “Numerical Approximation in CFD Problems Using Physics Informed Machine Learning” (Rout et al., 2021) uses DPINN to mean Distributed Physics Informed Neural Network, not discontinuity-aware in name. Yet the thesis explicitly applies domain splitting, interface matching, and subdomain normalization to advection-dominated regimes with sharp layers and square-pulse transport. A plausible implication is that distributed PINNs form one branch of discontinuity-aware methodology even when the acronym originally referred to domain distribution rather than shock awareness.

2. Why standard PINNs fail in discontinuous and sharp-gradient regimes

The central issue is the mismatch between smooth neural approximants and low-regularity PDE solutions. Classical PINNs enforce differential operators pointwise through automatic differentiation, but shocks and contacts satisfy the governing equations only in a weak sense. For Burgers and Euler equations, the conservative and non-conservative forms are equivalent for smooth solutions but not for discontinuous ones; Rankine–Hugoniot theory applies to conservative form, and non-conservative forms generally do not select the correct shock speed without special treatment (Neelan et al., 27 Jun 2025). Standard PINNs trained on strong-form residuals therefore face large residual spikes near discontinuities, optimization instability, and shock smearing.

Multiple papers isolate different manifestations of this pathology. In advection-dominated steady advection–diffusion, the solution is almost flat in most of the domain and drops sharply near the outflow; Lagaris-style trial-function PINNs fail for strongly advective cases, and Raissi-style PINNs only extend the solvable range modestly (Rout et al., 2021). In discontinuous 1D advection, Fourier features and adaptive loss weighting are introduced specifically because standard PINNs oversmooth jumps and exhibit spectral bias (Khosravi et al., 28 Jan 2026). In Burgers, Euler, and relativistic hydrodynamics, residual-only PINNs suffer from convergence issues and reduced shock accuracy, which motivates locally linearized or Roe-based modifications of the residual operator (Urbán et al., 13 Jun 2025). In Stokes interface problems, pressure is discontinuous and velocity derivatives jump across the interface, so a single smooth input representation is insufficient; the paper instead augments the inputs differently for pressure and velocity (Tseng et al., 2023).

A recurring explanation is spectral bias. High-frequency or near-discontinuous content is learned slowly, while low-frequency smooth components dominate early optimization. The discontinuity-aware KAN-based DPINN addresses this with adaptive Fourier-feature embedding and a discontinuity-aware Kolmogorov–Arnold network (Lei et al., 11 Jul 2025). The three-phase-flow DPINN likewise uses Fourier embedding and residual-adaptive blocks to mitigate spectral bias at thin interfaces (Lei et al., 28 Nov 2025). This suggests that discontinuity awareness is as much an optimization problem as a representational one.

3. Core design patterns in discontinuity-aware PINNs

Across the cited works, several recurring mechanisms define the DPINN landscape.

Domain decomposition and local approximation: In the distributed DPINN of (Rout et al., 2021), the domain is partitioned into blocks, each with its own small neural network. Interface losses enforce continuity of value and slope, and optionally flux. The method is motivated by the statement that “Each network has to approximate a simpler function.” By localizing steep gradients to a few blocks and normalizing each subdomain to $[0,1]$, the method handles advection-dominated boundary layers much better than a single global network.

Residual reweighting near shocks: The weighted-equation PINN of (Liu et al., 2022) introduces a gradient-dependent factor

$$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$

or the 1D analogue using $u_x$, so that residuals in compressive regions are downweighted. The stated goal is to “weaken the expression of the network near discontinuities” and let the network focus on smooth regions. The adaptive-weight-and-viscosity PINN of (Neelan et al., 27 Jun 2025) uses an analogous compression-aware factor

$d = 0.12(|u_x|-u_x)+1$

for Euler, dividing the residual by $d$ so that strongly compressive regions contribute less to the loss.

Artificial viscosity, but learned and localized: In (Neelan et al., 27 Jun 2025), a trainable artificial viscosity field $\nu(x,t)$ is penalized through $\mathcal{L}_\nu$ and inserted into Burgers and Euler residuals. In (Lei et al., 11 Jul 2025), the KAN-based DPINN uses a learnable local artificial viscosity $\mu(\mathbf{x})=\nu\,r(\mathbf{x})\,s(\mathbf{x})$, where $r$ is a spectral radius and $s$ is a shock sensor. In (Lei et al., 28 Nov 2025), the phase-field DPINN uses $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$0, where $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$1 is an interface sensor and $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$2 is a single trainable scalar. These constructions share the same intent: stabilize training near discontinuities while limiting diffusion away from them.

Weak-form and jump-aware physics losses: The hybrid DG–NN method of (Shen et al., 22 Aug 2025) replaces strong-form residuals by a DG-inspired RK2 weak-form loss and adds an explicit Rankine–Hugoniot loss

$$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$3

This directly enforces jump conditions at shocks instead of forcing a pointwise PDE where derivatives are undefined. The hybrid discrete-time PINN of (Lv et al., 2021) uses a discontinuity indicator to switch between automatic differentiation in smooth regions and WENO-Z evaluation of the convective derivative in nonsmooth regions.

Interface-aware feature engineering: For discontinuous media in Maxwell problems, (Nohra et al., 2024) enriches the network input with a level-set function $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$4, multiple smoothed Heaviside channels $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$5, and $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$6. For Stokes interface problems, (Tseng et al., 2023) uses two subnetworks with different augmented inputs: an indicator-function input for discontinuous pressure and a cusp-enforced level-set input for velocity derivative jumps. A plausible implication is that, for interface problems, discontinuity awareness often enters through geometry-aware coordinates rather than through PDE residual modification alone.

4. Representative formulations and architectures

The distributed DPINN of (Rout et al., 2021) provides a canonical piecewise-strong-form construction. For steady 1D advection–diffusion,

$$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$7

each block has a local network $$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$8, and the total loss is

$$\lambda = \frac{1}{\varepsilon_2\big(|\nabla\cdot \vec u| - \nabla\cdot \vec u \big) + 1},$$9

combining PDE residual, boundary loss, interface value matching, and interface slope matching. The thesis also discusses second-derivative matching and flux matching, concluding that second-derivative matching hurts convergence while flux matching is similar in effect to slope matching. For unsteady square-pulse advection, the domain is split in both space and time, and each block still uses shallow networks, underscoring that decomposition rather than width or depth is the main mechanism (Rout et al., 2021).

The KAN-based DPINN of (Lei et al., 11 Jul 2025) adopts a standard PINN backbone but inserts an adaptive Fourier-feature layer,

$u_x$0

with learnable Fourier matrix $u_x$1, followed by a discontinuity-aware KAN whose scalar activation is

$u_x$2

The dynamic tanh term is intended to capture steep shock-like behavior, while the spline term provides smooth approximation elsewhere (Lei et al., 11 Jul 2025).

The three-phase-flow DPINN of (Lei et al., 28 Nov 2025) uses Fourier embeddings,

$u_x$3

then constructs “discontinuity-aware residual-adaptive blocks” with DyT activations and trainable residual mixing coefficients. The total loss combines Allen–Cahn, Cahn–Hilliard, Navier–Stokes, initial-condition, and boundary-condition terms with dynamic balancing based on gradient norms (Lei et al., 28 Nov 2025).

The DG-based discontinuity-aware framework of (Shen et al., 22 Aug 2025) is structurally different. It uses a weak-form DG loss based on RK2 time stepping, a jump detector with residual weight

$u_x$4

and temporal progressive learning in which separate neural “columns” are trained on successive time subintervals. Earlier columns are frozen and provide pseudo-labels and lateral connections, which the authors argue reduces long-time error accumulation and improves temporal coherence (Shen et al., 22 Aug 2025).

The interface-problem literature adds yet another architectural pattern. In Maxwell problems, a single PINN is augmented with level-set-derived features and smoothed Heaviside material coefficients (Nohra et al., 2024). In Stokes interface problems, a “discontinuity and cusp capturing” PINN uses two subnetworks—one for pressure, one for velocity—with different interface-feature augmentations (Tseng et al., 2023). This reinforces the view that discontinuity awareness is problem-structured rather than monolithic.

5. Training strategies, hyperparameters, and empirical behavior

Several training patterns recur across the papers.

Two-stage optimization is common. The discontinuous-advection PINN of (Khosravi et al., 28 Jan 2026) first optimizes the Fourier-feature parameters while freezing the MLP, then freezes the Fourier features and trains the network weights. During the first stage, the loss term associated with the discontinuous component, such as $u_x$5, is assigned a larger weight. During the second stage, adaptive loss balancing is used to equalize gradient magnitudes across PDE, IC, and BC terms. This suggests that separating feature learning from solution fitting can be especially beneficial when discontinuities dominate the early loss landscape.

Adam followed by L-BFGS is another standard schedule. The adaptive-weight-and-viscosity PINN for Burgers and Euler uses Adam and then L-BFGS (Neelan et al., 27 Jun 2025). The DG–NN method uses 20k Adam iterations followed by 1k L-BFGS iterations for Burgers, with analogous two-phase training for Euler (Shen et al., 22 Aug 2025). The phase-field DPINN uses AdamW with cosine annealing and adaptive time-marching across subintervals (Lei et al., 28 Nov 2025). The KAN-based DPINN uses Adam for Burgers and L-BFGS for airfoil Euler problems, with markedly different loss weights between PDE and boundary terms in the airfoil setting (Lei et al., 11 Jul 2025).

Curriculum mechanisms appear in multiple forms. The electromagnetic I-PINN progressively increases the Heaviside sharpness parameter $u_x$6, starting from a smoother material transition and then sharpening the interface while warm-starting from previous weights (Nohra et al., 2024). The phase-field DPINN uses adaptive time-marching across temporal windows (Lei et al., 28 Nov 2025). The DG–NN model uses temporal progressive learning and pseudo-label supervision between tasks (Shen et al., 22 Aug 2025). A plausible implication is that discontinuity-aware training often benefits from staging either geometry sharpness, temporal horizon, or representation bandwidth.

The distributed DPINN thesis provides unusually explicit optimizer and hyperparameter comparisons. For shallow two-neuron-per-block networks, L-BFGS-B gave the best convergence, Adam was second, and tanh performed best among tested activations (Rout et al., 2021). The thesis also reports that increasing blocks in both space and time reduces loss in the square-pulse test, while too many collocation points per block can eventually lead to ill-conditioning. It further notes that PDE loss often dominates in highly advective cases and that increasing boundary and interface weights by factors of $u_x$7–$u_x$8 can restore balance (Rout et al., 2021).

6. Reported results, capabilities, and limitations

The distributed DPINN of (Rout et al., 2021) shows a large extension of solvable advection-dominated regimes. Classical Lagaris-style PINNs work roughly for $u_x$9, Raissi-style PINNs for $d = 0.12(|u_x|-u_x)+1$0, while DPINN with subdomain normalization and many blocks handles $d = 0.12(|u_x|-u_x)+1$1 down to $d = 0.12(|u_x|-u_x)+1$2 or even $d = 0.12(|u_x|-u_x)+1$3. The thesis describes this as a two-orders-of-magnitude improvement in the practically reachable advection-dominant limit. It also reports that an ELM-based DPINN variant is “non-iterative; solution in a single shot” and can solve advection–diffusion down to $d = 0.12(|u_x|-u_x)+1$4–0.003 with high accuracy (Rout et al., 2021).

The adaptive-weight-and-viscosity PINN of (Neelan et al., 27 Jun 2025) reports that for Burgers and Euler, conservative-form and non-conservative-form PINNs-AWV produce essentially the same shock solutions when the adaptive viscosity and residual weighting are used. For the Sod tube, the authors state that “PINNs‑AWV solution is independent of the form of the governing equation used.” They also report that pure adaptive-weight PINNs can obtain the correct shock speed in some cases but may fail to resolve the contact discontinuity, whereas adaptive-viscosity variants perform better (Neelan et al., 27 Jun 2025).

The DG-based discontinuity-aware method of (Shen et al., 22 Aug 2025) reports systematic gains over vanilla PINNs and competitive or better performance than first-order DG and PINNs-WE on Burgers and Euler. For Sod with 256 cells and two temporal tasks, their table gives $d = 0.12(|u_x|-u_x)+1$5, $d = 0.12(|u_x|-u_x)+1$6, and $d = 0.12(|u_x|-u_x)+1$7 errors of $d = 0.12(|u_x|-u_x)+1$8, $d = 0.12(|u_x|-u_x)+1$9, and $d$0, compared with $d$1, $d$2, and $d$3 for a vanilla PINN (Shen et al., 22 Aug 2025). The paper also presents theoretical bounds linking the NN approximation to the DG solution and then the DG solution to the entropy solution.

The KAN-based DPINN of (Lei et al., 11 Jul 2025) reports strong shock-resolution gains with far fewer parameters than MLP baselines. For inviscid Burgers, their table lists parameter counts of 744 for DPINN versus 4142 for the MLP baselines, with $d$4 error $d$5 for DPINN versus $d$6 for MLP + local artificial viscosity. For transonic NACA0012 with $d$7, the reported $d$8 error is $d$9 for DPINN versus $\nu(x,t)$0 for MLP + local AV and $\nu(x,t)$1 for plain MLP. For supersonic $\nu(x,t)$2, DPINN reaches $\nu(x,t)$3 versus $\nu(x,t)$4 for MLP + local AV (Lei et al., 11 Jul 2025).

The phase-field DPINN of (Lei et al., 28 Nov 2025) reports that standard PINNs fail as the interface thickness $\nu(x,t)$5 decreases, whereas DPINN improves. In the reversed single-vortex test, the paper gives $\nu(x,t)$6 for a standard PINN at $\nu(x,t)$7 versus $\nu(x,t)$8 for DPINN. In the bubble-rising benchmark, the center-of-mass error at $\nu(x,t)$9 is $\mathcal{L}_\nu$0 for PINN and $\mathcal{L}_\nu$1 for DPINN. The same method is then extended to a three-phase droplet-icing problem with viscosity and density contrasts exceeding 7 and 3 orders of magnitude, respectively (Lei et al., 28 Nov 2025).

Limitations are equally explicit. Domain decomposition requires empirical tuning of block numbers, collocation counts, and loss weights (Rout et al., 2021). Artificial-viscosity methods modify the PDE and may still smear contacts or depend on sensor quality (Neelan et al., 27 Jun 2025, Lei et al., 11 Jul 2025). Roe-based PINNs preserve shocks sharply but smooth out small-scale vortical structures in 2D interaction regions (Urbán et al., 13 Jun 2025). Phase-field DPINNs remain expensive because Allen–Cahn and Navier–Stokes need second derivatives and Cahn–Hilliard needs fourth derivatives (Lei et al., 28 Nov 2025). The DG–NN method notes that over-segmentation in time can degrade performance, with two or three tasks often better than four (Shen et al., 22 Aug 2025).

7. Relation to adjacent methods and evolving directions

The discontinuity-aware PINN literature sits at the intersection of several older strands: domain-decomposition PINNs, weak-form PINNs, entropy-aware PINNs, shock-capturing numerical schemes, and interface-enriched approximation spaces.

One branch uses decomposition and interface constraints. Distributed PINNs (Rout et al., 2021), XPINN/cPINN-like approaches mentioned in (Neelan et al., 27 Jun 2025), and separation–transfer PINNs (Wang et al., 26 May 2025) all reduce the complexity of learning multiple discontinuities by breaking the domain into subdomains or stages. ST-PINNs sequentially detect the strongest discontinuity using an intensity measure

$\mathcal{L}_\nu$2

then split the domain and retrain a new PINN via transfer learning on the remaining subproblem. For a 2D shock-refraction case, the paper reports reductions from $\mathcal{L}_\nu$3 to $\mathcal{L}_\nu$4 in $\mathcal{L}_\nu$5 and from $\mathcal{L}_\nu$6 to $\mathcal{L}_\nu$7 in $\mathcal{L}_\nu$8 when comparing GA-PINNs to ST-PINNs (Wang et al., 26 May 2025). This suggests a curriculum-on-discontinuities viewpoint complementary to explicit shock-aware residuals.

A second branch embeds discontinuity handling through discrete numerical structure. The hybrid PINN of (Lv et al., 2021) uses a discontinuity indicator to switch from AD to WENO-Z in nonsmooth regions. The DG-based method (Shen et al., 22 Aug 2025) builds the entire loss from DG residuals and Rankine–Hugoniot conditions. The approximate-Riemann-solver PINNs of (Urbán et al., 13 Jun 2025) locally replace the Jacobian by shock-speed-modified or Roe-averaged versions to enforce jump physics and conservation. These methods are the closest to classical CFD shock-capturing ideas.

A third branch relies on geometric feature enrichment for interface problems. Maxwell (Nohra et al., 2024) and Stokes (Tseng et al., 2023) problems do not primarily involve shocks, but they do involve discontinuous coefficients, discontinuous pressure, and derivative jumps. In these settings, discontinuity awareness comes from level-set features, Heaviside-type inputs, cusp-enforced enrichments, and strong imposition of boundary conditions. This broadens the term DPINN beyond hyperbolic conservation laws.

Recent work also shows that discontinuity-aware PINNs are beginning to separate from strict conservative-form dependence. PINNs-AWV reports that with adaptive viscosity and adaptive weighting, conservative and non-conservative formulations produce essentially the same shock structures for Burgers and Euler (Neelan et al., 27 Jun 2025). A plausible implication is that PINNs can partially decouple shock accuracy from conservative discretization if weak-solution regularization is built into the network and loss design. By contrast, the DG-based and Roe-based works indicate that explicit conservation-aware structures remain highly effective for shock-dominated flows (Shen et al., 22 Aug 2025, Urbán et al., 13 Jun 2025).

Taken together, the literature points toward a composite view of DPINN. A mature discontinuity-aware PINN is likely to combine several ingredients rather than only one: localized approximation or temporal decomposition, residual reweighting near singular regions, learned or numerical diffusion localized by sensors, weak-form or jump-based losses, and geometry-aware or frequency-aware feature enrichment. This suggests that the future of DPINN research will likely be hybrid rather than purely architectural or purely variational.