NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures
NEXT stabilizes physics-informed neuro-spectral models on stiff PDEs by integrating the linear part exactly.
PINNs suffer spectral bias and weak causality, while Neuro-Spectral Architectures reduce those issues but become unstable on stiff PDEs. NEXT combines NeuSA's spectral solution representation with high-order exponential integrators, integrating the stiff linear term exactly through matrix exponentials and modeling the nonlinear remainder with a neural network. Benchmarks show NEXT stays stable and accurate where NeuSA diverges numerically. It also handles inverse problems that recover unknown parameters or boundary conditions from sparse data, with code released publicly.
- Pairs NeuSA spectral PDE representations with exponential integrators.
- Stiff linear dynamics are integrated via matrix exponentials.
- Stable on stiff PDE benchmarks where NeuSA diverges.
- Also learns unknown parameters from sparse inverse-problem data.
Full article191 words · extracted from arxiv.org · click to collapse
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 .
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.31539