Stable neural method improves solving stiff physics equations
NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures
Machine Learning
Summary
Some computer methods struggle to solve physics problems involving certain tricky equations known as stiff PDEs because these problems are hard to compute accurately and stably. The authors propose a new approach called NEXT that combines a special neural network way to represent solutions with advanced math techniques to handle the tough parts exactly. This makes their method more stable and accurate on difficult physics problems and also lets it learn unknown parameters from limited data. Their tests showed that NEXT outperforms previous related methods that became unstable.
What this means in practice
- •For computational physicists: Run stable and accurate neural simulations for stiff physics PDEs that previously caused numerical failures.
- •For environmental modelers: Estimate unknown environmental parameters or boundary conditions from sparse measurement data using neural PDE models.
Authors
Márcio Marques, Leonardo Mendonça, Leonardo M. Moreira, Christian Júnior de Oliveira, Vitor Balestro, Tiago Novello, Daniel Yukimura, Pavel Petrov, Lucas Nissenbaum
Abstract
Physics-Informed Neural Networks (PINNs) build neural representations of time-dependent PDE solutions, naturally incorporating physics knowledge and observational data, which makes them well suited to both forward and inverse PDE problems. PINNs, however, are known to suffer from spectral bias and lack of causality. Neuro-Spectral Architectures (NeuSA), a recently proposed alternative to PINNs, mitigate both issues, but their numerical integration becomes unstable for stiff differential equations arising in many relevant physical problems. This study proposes Neuro-Spectral Exponential Time Differencing Architectures (NEXT), which combines the spectral representation of the PDE solution in NeuSA with high-order exponential integrators. Within this approach, the linear stiff part of the vector field induced by the PDE is integrated exactly through matrix exponentials, while the possibly nonlinear remainder is modeled by a neural network. The effectiveness of NEXT is verified through benchmark experiments on a set of stiff PDEs, in which NEXT is stable and accurate while NeuSA diverges numerically. It is also shown that NEXT can be applied to inverse problems, where the model has to learn unknown parameters or boundary conditions from sparse data. All code used in this work is publicly available at: https://github.com/marcioh2m/next.git .