# Competitive Physics Informed Networks

arXiv: https://arxiv.org/abs/2204.11144 · HTML: https://mlsanity.com/paper/2204.11144/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 58 references that are in the map.
- First posted on arXiv 23 Apr 2022.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [The Numerics of GANs](https://mlsanity.com/paper/1705.10461/) (2017)
- [Constrained Convolutional Neural Networks for Weakly Supervised Segmentation](https://mlsanity.com/paper/1506.03648/) (2015)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Simulation Intelligence: Towards a New Generation of Scientific Methods](https://mlsanity.com/paper/2112.03235/) (2021)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Competitive Gradient Descent](https://mlsanity.com/paper/1905.12103/) (2019)
- [FairALM: Augmented Lagrangian Method for Training Fair Models with Little Regret](https://mlsanity.com/paper/2004.01355/) (2020)
- [Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games](https://mlsanity.com/paper/1503.03467/) (2015)
- [Global Convergence to the Equilibrium of GANs using Variational Inequalities](https://mlsanity.com/paper/1808.01531/) (2018)
- [Implicit competitive regularization in GANs](https://mlsanity.com/paper/1910.05852/) (2019)
- [Weak Form Theory-guided Neural Network (TgNN-wf) for Deep Learning of Subsurface Single and Two-phase Flow](https://mlsanity.com/paper/2009.04543/) (2020)
- [Physics-Augmented Learning: A New Paradigm Beyond Physics-Informed Learning](https://mlsanity.com/paper/2109.13901/) (2021)
- [Competitive Mirror Descent](https://mlsanity.com/paper/2006.10179/) (2020)

## Built on by (most cited first, 19 of 19)

- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs](https://mlsanity.com/paper/2305.18702/) (2023)
- [Flow Factorized Representation Learning](https://mlsanity.com/paper/2309.13167/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Achieving High Accuracy with PINNs via Energy Natural Gradients](https://mlsanity.com/paper/2302.13163/) (2023)
- [ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learning](https://mlsanity.com/paper/2412.10782/) (2024)
- [Space and Time Continuous Physics Simulation From Partial Observations](https://mlsanity.com/paper/2401.09198/) (2024)
- [Dual Natural Gradient Descent for Scalable Training of Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.21404/) (2025)
- [OXYGENERATOR: Reconstructing Global Ocean Deoxygenation Over a Century with Deep Learning](https://mlsanity.com/paper/2405.07233/) (2024)
- [Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees](https://mlsanity.com/paper/2410.18153/) (2024)
- [Auxiliary-Tasks Learning for Physics-Informed Neural Network-Based Partial Differential Equations Solving](https://mlsanity.com/paper/2307.06167/) (2023)
- [Unsupervised Representation Learning from Sparse Transformation Analysis](https://mlsanity.com/paper/2410.05564/) (2024)
- [A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations](https://mlsanity.com/paper/2411.06278/) (2024)
- [Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra](https://mlsanity.com/paper/2505.11638/) (2025)
- [A Residual Guided strategy with Generative Adversarial Networks in training Physics-Informed Transformer Networks](https://mlsanity.com/paper/2508.00855/) (2025)
- [Integrating Artificial Intelligence, Physics, and Internet of Things: A Framework for Cultural Heritage Conservation](https://mlsanity.com/paper/2604.03233/) (2026)
- [Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers](https://mlsanity.com/paper/2607.22004/) (2026)

## Abstract (arXiv)

Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy solutions, typically attaining about 0.1% relative error. We present an adversarial approach that overcomes this limitation, which we call competitive PINNs (CPINNs). CPINNs train a discriminator that is rewarded for predicting mistakes the PINN makes. The discriminator and PINN participate in a zero-sum game with the exact PDE solution as an optimal strategy. This approach avoids squaring the large condition numbers of PDE discretizations, which is the likely reason for failures of previous attempts to decrease PINN errors even on benign problems. Numerical experiments on a Poisson problem show that CPINNs achieve errors four orders of magnitude smaller than the best-performing PINN. We observe relative errors on the order of single-precision accuracy, consistently decreasing with each epoch. To the authors' knowledge, this is the first time this level of accuracy and convergence behavior has been achieved. Additional experiments on the nonlinear Schr"odinger, Burgers', and Allen-Cahn equation show that the benefits of CPINNs are not limited to linear problems.