aiqbench analysis
This research introduces NEXT, a novel architecture that merges Neuro-Spectral methods with exponential time differencing to solve stiff partial differential equations more stably than existing models. By integrating high-order matrix exponentials for linear components and neural networks for nonlinear remainders, the approach effectively addresses spectral bias and causality issues inherent in standard Physics-Informed Neural Networks. The study demonstrates significant improvements in accuracy and stability for both forward and inverse problems involving sparse data.
NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures
arXiv:2609.31539v1 Announce Type: new 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 .
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: this https URL .
Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2609.31539 [cs.LG]
(or arXiv:2609.31539v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.31539 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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