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Boundary-Informed Method of Lines for Physics Informed Neural Networks

Maximilian Cederholm, Siyao Wang, Haochun Wang, Ruichen Xu, Yuefan Deng

TL;DR

This work introduces boundary-informed MOL-PINN, a hybrid solver that unifies the dimensionality-reduction strength of the Method of Lines with the differentiability of Physics-Informed Neural Networks. A two-stage process learns a spatial surrogate at $t=0$, enabling spectrally accurate spatial derivatives via automatic differentiation, and then evolves in time within a reduced space, either through classical integrators or a temporal PINN. The method achieves high spatial accuracy with far fewer residual evaluations than traditional PINNs or FD-based MOL, reduces memory requirements, and extends naturally to linear and nonlinear PDEs in multiple dimensions. While demonstrating strong performance and robustness, it also highlights computational trade-offs at large system counts and outlines concrete directions for adaptive refinement and multi-directional MOL extensions.

Abstract

We propose a hybrid solver that fuses the dimensionality-reduction strengths of the Method of Lines (MOL) with the flexibility of Physics-Informed Neural Networks (PINNs). Instead of approximating spatial derivatives with fixed finite-difference stencils - whose truncation errors force extremely fine meshes - our method trains a neural network to represent the initial spatial profile and then employs automatic differentiation to obtain spectrally accurate gradients at arbitrary nodes. These high-fidelity derivatives define the right-hand side of the MOL-generated ordinary-differential system, and time integration is replaced with a secondary temporal PINN while spatial accuracy is retained without mesh refinement. The resulting "boundary-informed MOL-PINN" matches or surpasses conventional MOL in accuracy using an order of magnitude fewer collocation points, thereby shrinking memory footprints, lessening dependence on large data sets, and increasing complexity robustness. Because it relies only on automatic differentiation and standard optimizers, the framework extends naturally to linear and nonlinear PDEs in any spatial dimension.

Boundary-Informed Method of Lines for Physics Informed Neural Networks

TL;DR

This work introduces boundary-informed MOL-PINN, a hybrid solver that unifies the dimensionality-reduction strength of the Method of Lines with the differentiability of Physics-Informed Neural Networks. A two-stage process learns a spatial surrogate at , enabling spectrally accurate spatial derivatives via automatic differentiation, and then evolves in time within a reduced space, either through classical integrators or a temporal PINN. The method achieves high spatial accuracy with far fewer residual evaluations than traditional PINNs or FD-based MOL, reduces memory requirements, and extends naturally to linear and nonlinear PDEs in multiple dimensions. While demonstrating strong performance and robustness, it also highlights computational trade-offs at large system counts and outlines concrete directions for adaptive refinement and multi-directional MOL extensions.

Abstract

We propose a hybrid solver that fuses the dimensionality-reduction strengths of the Method of Lines (MOL) with the flexibility of Physics-Informed Neural Networks (PINNs). Instead of approximating spatial derivatives with fixed finite-difference stencils - whose truncation errors force extremely fine meshes - our method trains a neural network to represent the initial spatial profile and then employs automatic differentiation to obtain spectrally accurate gradients at arbitrary nodes. These high-fidelity derivatives define the right-hand side of the MOL-generated ordinary-differential system, and time integration is replaced with a secondary temporal PINN while spatial accuracy is retained without mesh refinement. The resulting "boundary-informed MOL-PINN" matches or surpasses conventional MOL in accuracy using an order of magnitude fewer collocation points, thereby shrinking memory footprints, lessening dependence on large data sets, and increasing complexity robustness. Because it relies only on automatic differentiation and standard optimizers, the framework extends naturally to linear and nonlinear PDEs in any spatial dimension.
Paper Structure (13 sections, 5 equations, 4 figures, 4 tables)

This paper contains 13 sections, 5 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries.
  3. Method of Lines (MOL).
  4. Neural networks.
  5. Physics‑Informed Neural Networks (PINNs).
  6. From MOL to Boundary‑informed MOL‑PINN.
  7. Stage I — Spatial surrogate learning.
  8. Stage II — Temporal evolution in reduced space.
  9. Hardware and Hyper-parameters
  10. Results
  11. N-S Results
  12. Time-based Efficiency
  13. Conclusion and future work.

Figures (4)

  • Figure 1: Visual comparison between PINN and MOL-PINN architectures.
  • Figure 2: Image 1 caption
  • Figure 3: Image 2 caption
  • Figure 4: Image 3 caption