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Gen-PINNs: Generative Adversarial Physics Informed Neural Networks for solving partial differential equations

Muhammad M. Akmal, Kamy Sepehrnoori, Michael J. Pyrcz

arXiv:2610.10897v1cs.LGphysics.comp-phphysics.flu-dyn

TL;DR

PINN extensions often address individual PDE modeling challenges, while methods may struggle to maintain accuracy as solution characteristics vary. Gen-PINNs combines deterministic residual-adversarial learning with representation and optimization strategies, and the paper reports improved accuracy and convergence across challenging PDE regimes.

  • Problem

    Existing PINN extensions often target individual difficulties and may struggle to maintain accuracy when PDE solution characteristics vary.

    Source: §1 · p. 3

  • Method

    Gen-PINNs uses a deterministic coordinate-conditioned generator and multiple discriminators that evaluate complementary PDE residual features against ideal zero-residual states.

    Source: §1 · p. 3

  • Results

    Gen-PINNs improves accuracy and convergence over standard PINNs across nonlinear, stiff, sharp-front, and higher-order PDEs considered in the study.

    Source: §5 · pp. 40–41

  • Takeaways & Limitations

    The results support the potential of deterministic residual-adversarial learning for physics-informed PDE solutions across the evaluated regimes.

    Source: §5 · p. 41

    Standard PINNs struggle to converge sufficiently as PDE complexity grows, with spectral bias, fixed loss weights, and first-order optimizers among the stated constraints.

    Source: §2.2 · p. 12

Abstract

from arXiv · show

Physics-Informed Neural Networks (PINNs) are a widely used data-free method for solving Partial Differential Equations (PDEs) using machine learning. With recent advances in Generative Adversarial Networks (GANs), adversarial learning has shown strong capabilities for modeling complex data-driven problems; however, the use of GANs in deterministic physics-informed PDE solutions remains limited. In this work, we first identify limitations of standard PINNs for solving PDEs, including spectral bias, loss imbalance, and optimizer stagnation. We then propose Generative Adversarial Physics-Informed Neural Networks (Gen-PINNs), a unified deterministic residual-adversarial framework designed to improve data-free solutions of PDEs with sharp or shock-front behavior. The generator learns the underlying PDE solution using dynamically weighted physics-informed loss components, while separate discriminators evaluate complementary PDE residual features against ideal zero-residual states. The framework further develops and adapts several methodological components, including a Fourier representation for resolving high-frequency spatial content, an orthonormal spectral diagnostic for quantifying frequency-dependent solution errors, and a modified gradient-based dynamic weighting system for physics, initial-condition, boundary-condition, and adversarial loss objectives. Gen-PINNs is tested against standard PINNs on nonlinear and higher-order PDEs, including the Burgers, Allen-Cahn, and Kuramoto-Sivashinsky equations. The results demonstrate substantial improvements in accuracy and convergence across sharp-front, stiff, and higher-order PDE solutions, highlighting the potential of deterministic residual-adversarial learning as an effective approach for solving challenging nonlinear PDEs.

1. Introduction

PDEs are important across science and engineering, but existing PINN variants often address individual modeling challenges without generalizing across changing solution characteristics. Gen-PINNs combines deterministic residual-adversarial learning with representation and optimization strategies, and the paper reports improved accuracy and convergence across challenging PDE regimes.

1. Introduction

  • Standard PINNs can exhibit training stiffness, converge to trivial solutions for shock-front PDEs, and favor low-frequency over high-frequency solution regions.Training stiffness is described as small parameter changes causing large changes in PDE solution error.Source: §1 · p. 2
  • Existing PINN extensions often target individual challenges, leaving a need for approaches that address multiple PDE modeling difficulties together.The paper identifies longer training times, complex loss landscapes, poor optimizer performance, and hyperparameter sensitivity among the challenges associated with these extensions.Source: §1 · p. 2
  • PINN extensions may lose accuracy when solution characteristics vary, motivating methods that adapt across PDEs, initial or boundary conditions, and coefficients.The paper notes that many extensions perform well only for particular PDEs at fixed conditions.Source: §1 · pp. 2–3
  • GANs have modeled sharp discontinuities, oscillations, and multiscale structures, but GAN-based PINN extensions remain scarce.Prior examples include physics-informed GANs for stochastic PDEs and other hybrid or uncertainty-oriented extensions.Source: §1 · p. 3
  • Gen-PINNs improves accuracy and convergence over standard PINNs across nonlinear, stiff, sharp-front, and higher-order PDE configurations.The framework combines a deterministic coordinate-conditioned generator with multiple discriminators that evaluate complementary physics residual features against zero-residual states.Source: §1 · p. 3

2. Physics Informed Neural Networks (PINNs)

PINNs approximate PDE solutions with neural networks trained against the PDE residual and initial and boundary conditions. The Burgers’ example illustrates their shock-front error and examines spectral bias, loss imbalance, optimizer stagnation, and uniform MSE limitations.

2.1. Problem Description

  • PINNs use a neural network to approximate the solution, minimizing a composite objective built from PDE residual, initial-condition, and boundary-condition losses.The PDE residual is evaluated at collocation points, while initial and boundary losses use their corresponding known data.Source: §2 · p. 4, §2.1 · pp. 4–5

2.2. Standard PINNs limitations

  • 35.41% average relative L2 error remained after 500 epochs, with a 59.6% maximum at x=-0.035, t=0.515.The PINN roughly captured the narrow shock front but struggled with the smooth trends on either side of the transition.Source: §2.2 · p. 6, Figure 1 · p. 7
  • The Dirichlet-compatible spectral diagnostic applies an orthonormal DST-1 basis to interior points and compares energy-weighted relative error across wavenumber bins.The resulting spectrum plots binned error against representative wavenumber.Source: §2.2 · pp. 8–9
  • As wavenumber increases, the relative spectral error rises, indicating that standard PINNs learn lower-frequency features before higher-frequency detail.The missed-energy fraction grows at higher wavenumbers, where complex solution details remain harder to resolve.Source: §2.2 · p. 9
  • Constant loss weights can leave PDE, initial, and boundary objectives converging at different rates, while first-order optimization can stall or overshoot in stiff, anisotropic loss landscapes.The Burgers diagnostics show higher-frequency errors persisting and PDE loss remaining relatively high as initial- and boundary-condition losses decrease.Source: §2.2 · pp. 10–11
  • Uniform MSE weighting treats collocation points equally, so shock-front and high-frequency regions may remain under-resolved alongside smoother regions.The paper identifies this loss behavior, together with spectral bias and optimization issues, as a convergence limitation as PDE complexity grows.Source: §2.2 · p. 12

3. Generative Adversarial Physics-Informed Neural Networks (Gen-PINNs)

This section describes Gen-PINNs’ deterministic residual-adversarial training, including its representation, loss, and optimization components and problem-informed configuration. It also reports improved Burgers accuracy and frequency resolution compared with standard PINNs.

3.1. Gen-PINNs Architecture

  • Gen-PINNs combines LS-GAN physics-based adversarial losses, Fourier embeddings, dynamic loss weighting, causal PDE loss, mixed optimizers, and FP64 training in one architecture.The components target representation and optimization limitations including spectral bias, loss imbalance, heterogeneous residual errors, and optimizer stagnation.Source: §3.1 · p. 13
  • Fourier input embeddings provide high-frequency features directly to the generator, helping it model fine-scale structures and reduce spectral bias.For the Burgers’ PDE case, the embedding improves modeling of the shock front at x=0; explicit boundary-condition loss still imposes prescribed Dirichlet values.Source: §3.1 · pp. 14–15
  • Conditioned on spatiotemporal coordinates, separate discriminators assess PDE, H1, signed cubic, and quartic residual features, each of which should approach zero for a physically accurate solution.The H1 feature penalizes residual roughness, the signed cubic feature preserves residual direction, and the quartic feature penalizes large residuals regardless of sign.Source: §3.1 · p. 16
  • A single generator spans the full temporal domain, with a tertiary causality-based PDE loss using time-bin weights and variance-informed MSE to emphasize earlier residuals.The framework includes stabilized non-differentiable and unrestricted differentiable causal-gradient pathways; the latter may improve convergence in some PDEs but can destabilize higher-order sharp-feature cases.Source: §3.1 · pp. 18–19
  • Modified gradient balancing dynamically adjusts IC, BC, PDE, adversarial, and causal-loss weights to limit domination by any single objective during training.The method compares each loss component’s gradient norm with the geometric mean of active norms; FP64 training, delayed discriminator updates, and a second-half L-BFGS stage are also included.Source: §3 · p. 13, §3.1 · pp. 14, 19
  • Gen-PINNs combines direct physics constraints, adaptive gradient-norm loss weighting, and residual-adversarial feedback; VIMSE applies mean-and-variance error weighting to active loss components.The adaptive weights balance generator gradient contributions across loss terms, while discriminators evaluate residual features and feed adversarial loss back to the generator.Source: §3.1 · pp. 20–21
  • Training uses ADAM followed by L-BFGS, with one discriminator update per 10 generator updates; dynamic weights retain 70% of the previous weight for smoother changes.The reported settings include an ADAM learning rate of 10^-3 and L-BFGS gradient tolerance of 10^-6.Source: §3.1 · pp. 20, 22

3.2. Hyperparameter tuning

  • Algorithm 1 configures Gen-PINNs by matching observed PDE and training characteristics to components; for Burgers’ PDE case 1, 10 Fourier features were specified and dynamic weights were not applied.Source: §3.2 · pp. 22–23

3.3. Gen-PINNs application to the Burgers’ benchmark PDE

  • Across 500 epochs, relative error fell from 35.41% in standard PINNs to 2.99% in Gen-PINNs, which matched the numerical Burgers shock front at x=0.The predicted solution also matched the numerical solution’s extrema and inflection points.Source: §3.3 · p. 25, Figure 5 · p. 26, Figure 4 · p. 26
  • Gen-PINNs’ Rbin began at 0.5×10^-2 versus 0.5 for standard PINNs, and ΔF(κ) fell below 0.1 for all wavenumbers.The higher modes’ errors decreased during training, while IC, BC, and PDE errors fell more than under standard PINNs in the same training conditions.Source: §3.3 · p. 25, Figure 6 · p. 26

4. PDE Experiment Results

The experiments evaluate Gen-PINNs against standard PINNs on one-dimensional forward PDEs, using numerical references and relative L2 error to assess accuracy across different challenges. Results vary by PDE and configuration, with ablations showing that effective components depend on temporal and spectral characteristics, and some gains accompanied by lower training throughput.

4.1. Burgers’ PDE case 2

  • At 100,000 iterations on Burgers’ case 2, relative L2 error was 0.77% for Gen-PINNs and 0.87% for standard PINNs, narrowing the short-budget gap.At 1,000 iterations, causal PDE loss plus dynamic weighting yielded 1.42% for Gen-PINNs versus 50.63% for standard PINNs; increasing auxiliary adversarial weights could worsen accuracy.Source: §4.1 · p. 28

4.2. Burgers’ PDE case 3

  • At 10,000 iterations, Gen-PINNs reached 0.015% relative L2 error versus 0.5% for standard PINNs on Burgers’ case 3, about a 30-fold improvement.The optimized configuration removed causal PDE loss, added random Fourier features in both x and t, and used only the PDE adversarial loss; it also matched the solution across time.Source: §4.2 · p. 30, Figure 9 · p. 31, Figure 10 · p. 31

4.3. Allen Cahn PDE

  • For Allen–Cahn, Gen-PINNs reached 2.06% relative L2 error, improving on the standard PINN’s 51.91% under the reported configurations.The final Gen-PINNs run used stabilized causal treatment and all adversarial PDE loss components at low auxiliary weights; Figures 12–13 show its predictions and error.Source: §4.3 · pp. 32–33, Figure 12 · p. 33, Figure 13 · p. 34

4.4. Kuramoto-Sivashinsky PDE

  • For Kuramoto–Sivashinsky, Gen-PINNs reached 2.52% relative L2 error versus 2.93% for standard PINNs at the same iteration count, mainly improving boundary modeling.The optimized configuration removed tertiary causal loss and retained a low-weight Sobolev-type adversarial feature; the single network still struggled with higher-order terms across the full time domain.Source: §4.4 · pp. 34–35, Figure 14 · p. 35, Figure 15 · p. 36

4.5 Gen-PINNs component ablation and sensitivity analysis

  • The component sensitivity analysis used targeted ablations rather than a full factorial study, finding that overweighting auxiliary residual features degrades convergence while longer optimization can improve accuracy.The analysis consolidated configuration changes and accuracy observations across PDE characteristics; the KS sensitivity study selected 15 configurations from hundreds of runs.Source: §4.5 · pp. 36–37, Table 2 · p. 36
  • Sensitivity results indicate that causal weighting can favor early-time errors and impede later propagation, while problem-informed reconfiguration removes this temporal bias.For strongly causal problems, removing the residual loss can allow the causal loss to improve the solution.Source: §4.5 · p. 37
  • Causal weighting improved Burgers case 2 but caused early-time bias that impeded later-time propagation in case 3, where conventional residuals and space-time Fourier embeddings improved the solution.The results indicate that causal weighting is a temporal optimization mechanism, not a universally beneficial loss component.Source: §4.5 · p. 37
  • For Allen–Cahn, removing the competing PDE residual and stabilizing the causal pathway reduced optimization stagnation and later-time errors; adversarial-feature loss then cut final relative error by more than half.The stabilized pathway achieved further error reduction with a single neural network over the full spatiotemporal domain.Source: §4.5 · p. 38
  • For KS, causal weighting hindered temporal propagation, and the best configuration removed causal PDE loss while retaining a small Sobolev-type adversarial contribution, marginally outperforming standard PINNs.Reducing causal-weight strength improved convergence but was insufficient to beat standard PINNs.Source: §4.5 · p. 38
  • JAX.jet minimally changed final error in KS, while the single-generator structure remained a limitation that may benefit from temporally decomposed multiple generators.The authors frame ablations as evidence of component behavior in the current framework, not proof of universal applicability or causal contribution.Source: §4.5 · p. 38

4.6. Performance comparison for Standard PINNs and Gen-PINNs

  • Gen-PINNs reduced relative L2 error across Burgers cases 1–3 and Allen–Cahn, but improved KS only slightly despite lower training throughput.Relative L2 error changed from 35.41% to 2.99% for Burgers case 1, 50.63% to 1.42% for case 2, 0.508% to 0.0147% for case 3, and 51.9% to 2.06% for Allen–Cahn; for KS it changed from 2.93% to 2.52%.Source: Table 3 · p. 38, §4.6 · p. 39

5. Conclusion

Gen-PINNs combines adversarial residual learning with supporting strategies to improve PDE-solution accuracy and convergence across the regimes studied. The conclusion also identifies configuration effort and computational cost as practical constraints.

5. Conclusion

  • The framework targets PINN training pathologies with residual-adversarial feedback and supporting strategies for representation, temporal modeling, weighting, and optimization.The identified problems include low-frequency spectral bias, fixed loss weights, and slow or oscillatory convergence with first-order optimizers and standard MSE.Source: §5 · p. 39
  • Gen-PINNs reduces relative L2 error by an average order of magnitude versus standard PINNs across studied regimes; Allen-Cahn errors are 51.9% versus 2.06%.The conclusion reports improvements across nonlinear, stiff, sharp-front, and higher-order PDEs.Source: §5 · p. 40
  • Gen-PINNs may require structured, problem-specific configuration, and unknown reference solutions make selecting suitable components more challenging.The authors note that automated component and hyperparameter selection remains future work to reduce configuration effort.Source: §5 · p. 40
  • The authors identify a unified loss for initial conditions, boundary conditions, and PDE residuals, Fourier-domain training, and multidimensional modeling as directions for further work.They also call for reducing the computational costs of the comprehensive optimization process.Source: §5 · pp. 40–41

CRediT authorship contribution statement

The contribution statement assigns methodological and analytical work across the three authors, with software, validation, and data curation listed for Akmal.

CRediT authorship contribution statement

  • Akmal is credited with methodology, investigation, formal analysis, software, visualization, validation, data curation, and writing contributions.The statement also credits Akmal with conceptualization.Source: CRediT authorship contribution statement · p. 41
  • Sepehrnoori is credited with methodology, investigation, formal analysis, visualization, supervision, project administration, funding acquisition, and writing contributions.The statement also credits Sepehrnoori with conceptualization.Source: CRediT authorship contribution statement · p. 41
  • Pyrcz is credited with methodology, investigation, formal analysis, visualization, supervision, project administration, funding acquisition, and writing contributions.The statement also credits Pyrcz with conceptualization.Source: CRediT authorship contribution statement · p. 41

Appendix

The appendix reports configuration-dependent outcomes across PDE experiments and analyzes how residual features and discriminator responses shape generator updates. It also describes when causal weighting, global PDE loss, and exponential weighting hinder or improve convergence.

Appendix

  • Squared, cubic, quartic, and H1 residual features expose generator updates to residual magnitude, sign, and spatial variation.The relative contributions of higher-order features are controlled by β_s multipliers.Source: §B.1 · p. 51
  • Discriminator responses dynamically weight adversarial gradients across space, focusing generator updates where physics-informed loss is higher.The discriminators receive coordinates, allowing the adversarial contribution to vary across collocation points.Source: §B.2 · p. 52
  • With causal PDE weights on, standard PINNs records 2.02% error while Gen-PINNs records 11%.The comparison is reported for the Burgers PDE configuration in which causal PDE weights are turned on.Source: §C.2 · p. 56
  • For Burgers with varying boundary conditions, Gen-PINNs errors rise over time under causal PDE loss, while standard PINNs errors remain relatively steady.The authors suggest the global PDE loss can sometimes suffice when varying boundary conditions strongly influence the solution.Source: §C.2 · p. 56
  • Adding learnable Fourier features in x and t improves the Burgers match with non-standard boundary conditions.The time dimension is added to the Fourier input embedding; the authors state that its mathematical implications are beyond the work’s scope.Source: Figure 22 · p. 57, §C.2 · p. 57
  • Using global and tertiary causal PDE losses simultaneously produces poor convergence; the authors recommend tertiary causal loss only for strongly causal PDEs.Turning off global PDE loss improves modeling, and a stabilized causal update scheme improves convergence for the Allen-Cahn case.Source: §C.2 · pp. 58–60
  • At α_exp=1 × 10², Gen-PINNs has a time cutoff and 45.3% relative L2 error; reducing α_exp to 1 × 10¹ yields 11.2%.Increasing training iterations beyond 10,000 only slightly improved the first configuration.Source: §C.2 · p. 61
  • For the Kuramoto–Sivashinsky equation, Gen-PINNs reaches 2.52% relative error versus 2.93% for standard PINNs.The authors note that removing the temporally weighted PDE objective gives the strongest solutions; they suggest the fourth-order derivative may be poorly represented at later times.Source: §F.2 · p. 62

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