mlsanityData to 7 Oct 2026

PIAT: Physics Informed Adversarial Training for Solving Partial Differential Equations

arXiv 2207.06647 ↗First posted 14 Jul 2022cs.LGcited by 3

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Abstract (arXiv)

In this paper, we propose the physics informed adversarial training (PIAT) of neural networks for solving nonlinear differential equations (NDE). It is well-known that the standard training of neural networks results in non-smooth functions. Adversarial training (AT) is an established defense mechanism against adversarial attacks, which could also help in making the solution smooth. AT include augmenting the training mini-batch with a perturbation that makes the network output mismatch the desired output adversarially. Unlike formal AT, which relies only on the training data, here we encode the governing physical laws in the form of nonlinear differential equations using automatic differentiation in the adversarial network architecture. We compare PIAT with PINN to indicate the effectiveness of our method in solving NDEs for up to 10 dimensions. Moreover, we propose weight decay and Gaussian smoothing to demonstrate the PIAT advantages. The code repository is available at https://github.com/rohban-lab/PIAT.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 24 of its 46 references that are in the map. First posted on arXiv 14 Jul 2022.

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24 of 46 references.

References · 24

  1. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,326
  2. 2013Intriguing properties of neural networkscited by 5,114
  3. 2019Theoretically Principled Trade-off between Robustness and Accuracycited by 1,510
  4. 2019Adversarial Examples Are Not Bugs, They Are Featurescited by 950
  5. 2014Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Imagescited by 774
  6. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  7. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  8. 2020Underspecification Presents Challenges for Credibility in Modern Machine Learningcited by 488
  9. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  10. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  11. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  12. 2020Attacks Which Do Not Kill Training Make Adversarial Learning Strongercited by 242
12 more
  1. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  2. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  3. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  4. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  5. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  6. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  7. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  8. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  9. 2017Improving Deep Learning using Generic Data Augmentationcited by 36
  10. 2018Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networkscited by 20
  11. 2018Predictive Collective Variable Discovery with Deep Bayesian Modelscited by 5
  12. 2021Lagrangian Objective Function Leads to Improved Unforeseen Attack Generalization in Adversarial Trainingcited by 3

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Cited by year

Citing papers per year: 2023: 3.

3 papers.

  1. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
  2. 2023Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron modelscited by 4
  3. 2023A Data-Centric Approach for Improving Adversarial Training Through the Lens of Out-of-Distribution Detectioncited by 0