Generative adversarial networks improve solving complex partial differential equations
Gen-PINNs: Generative Adversarial Physics Informed Neural Networks for solving partial differential equations
Machine Learning
Summary
Solving equations that describe physical phenomena, called partial differential equations, can be hard for computers, especially when sudden changes or sharp fronts appear. The authors identify problems in an existing method called Physics-Informed Neural Networks (PINNs) and design a new approach named Gen-PINNs that combines the strengths of GANs with physics-based learning. Their method uses a generator to predict solutions and discriminators to check accuracy, improving precision on tough cases. Tests on several nonlinear equations show better results and faster convergence compared to standard techniques.
What this means in practice
- •For computational fluid dynamics engineers: Increase accuracy in simulating fluid flows with shocks or sharp changes by using Gen-PINNs to solve complex PDEs more reliably.
- •For materials science modelers: Model phase transitions and interface dynamics more effectively by applying Gen-PINNs to solve PDEs with stiff or nonlinear behavior.
Authors
Muhammad M. Akmal, Kamy Sepehrnoori, Michael J. Pyrcz
Abstract
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.