Neuro-Spectral Architectures for Causal Physics-Informed Networks

Arthur Bizzi, Leonardo M. Moreira, Márcio Marques, Leonardo Mendonça, Christian Júnior de Oliveira, Vitor Balestro, Lucas dos Santos Fernandez, Daniel Yukimura, Pavel Petrov, João M. Pereira, Tiago Novello, Lucas Nissenbaum

Keywords: Physics-informed neural networks, spectral methods, causality

Abstract:

Physics-Informed Neural Networks (PINNs) have emerged as a powerful frame-
work for solving partial differential equations (PDEs). However, standard MLP-
based PINNs often fail to converge when dealing with complex initial value
problems, leading to solutions that violate causality and suffer from a spectral
bias towards low-frequency components. To address these issues, we introduce
NeuSA (Neuro-Spectral Architectures), a novel class of PINNs inspired by classi-
cal spectral methods, designed to solve linear and nonlinear PDEs with variable
coefficients. NeuSA learns a projection of the underlying PDE onto a spectral
basis, leading to a finite-dimensional representation of the dynamics which is then
integrated with an adapted Neural ODE (NODE). This allows us to overcome
spectral bias, by leveraging the high-frequency components enabled by the spectral
representation; to enforce causality, by inheriting the causal structure of NODEs,
and to start training near the target solution, by means of an initialization scheme
based on classical methods. We validate NeuSA on canonical benchmarks for lin-
ear and nonlinear wave equations, demonstrating strong performance as compared
to other architectures, with faster convergence, improved temporal consistency
and superior predictive accuracy. Code and pretrained models are available in
https://github.com/arthur-bizzi/neusa.

Primary Area: Machine learning for sciences (e.g. climate, health, life sciences, physics, social sciences)

Submission Number: 13846