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TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions

Hongjie Jiang, Di Luo

TL;DR

TENG-BC, a high-precision neural PDE solver based on the Time-Evolving Natural Gradient, designed to perform under general boundary constraints, achieves solver-level accuracy under comparable sampling budgets, outperforming conventional solvers and physics-informed neural network (PINN) baselines.

Abstract

Accurately solving time-dependent partial differential equations (PDEs) with neural networks remains challenging due to long-time error accumulation and the difficulty of enforcing general boundary conditions. We introduce TENG-BC, a high-precision neural PDE solver based on the Time-Evolving Natural Gradient, designed to perform under general boundary constraints. At each time step, TENG-BC performs a boundary-aware optimization that jointly enforces interior dynamics and boundary conditions, accommodating Dirichlet, Neumann, Robin, and mixed types within a unified framework. This formulation admits a natural-gradient interpretation, enabling stable time evolution without delicate penalty tuning. Across benchmarks over diffusion, transport, and nonlinear PDEs with various boundary conditions, TENG-BC achieves solver-level accuracy under comparable sampling budgets, outperforming conventional solvers and physics-informed neural network (PINN) baselines.

TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions

TL;DR

TENG-BC, a high-precision neural PDE solver based on the Time-Evolving Natural Gradient, designed to perform under general boundary constraints, achieves solver-level accuracy under comparable sampling budgets, outperforming conventional solvers and physics-informed neural network (PINN) baselines.

Abstract

Accurately solving time-dependent partial differential equations (PDEs) with neural networks remains challenging due to long-time error accumulation and the difficulty of enforcing general boundary conditions. We introduce TENG-BC, a high-precision neural PDE solver based on the Time-Evolving Natural Gradient, designed to perform under general boundary constraints. At each time step, TENG-BC performs a boundary-aware optimization that jointly enforces interior dynamics and boundary conditions, accommodating Dirichlet, Neumann, Robin, and mixed types within a unified framework. This formulation admits a natural-gradient interpretation, enabling stable time evolution without delicate penalty tuning. Across benchmarks over diffusion, transport, and nonlinear PDEs with various boundary conditions, TENG-BC achieves solver-level accuracy under comparable sampling budgets, outperforming conventional solvers and physics-informed neural network (PINN) baselines.
Paper Structure (42 sections, 55 equations, 14 figures, 2 tables, 1 algorithm)

This paper contains 42 sections, 55 equations, 14 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Neural solvers for PDEs
  5. Boundary treatment in neural PDE solvers
  6. Methods
  7. Problem Formulation
  8. Boundary-Aware Least-Squares Stepper
  9. TENG-BC: TENG with arbitrary boundaries
  10. Combined interior--boundary update.
  11. Temporal discretization.
  12. Implementation Details
  13. Results
  14. Experimental Setup
  15. Heat Equation with Different Boundaries
  16. ...and 27 more sections

Figures (14)

  • Figure 1: Overview of the boundary-aware TENG-BC update. The neural representation is advanced by solving a unified least-squares problem that enforces both interior dynamics and boundary constraints. Different boundary types are incorporated through the same optimization operator, which is repeatedly applied to realize temporal evolution.
  • Figure 2: Performance of TENG-BC on heat equation with different boundary conditions.(a) TENG-Heun predictions at $T=1.0$ are shown with analytic references and pointwise error maps. (b) Per-time-step relative $L^2$-error comparison among TENG-BC variants and other reference solvers.
  • Figure 3: Performance of TENG-BC on the two-dimensional transport equation with nonzero dirichlet boundary. Comparison of per-time-step relative $L^2$-error among TENG-BC variants, PINN, and FEM baselines.
  • Figure 4: Performance of TENG-BC on the Burgers equation with periodic boundary.(a) Predicted solutions by TENG-Heun at representative time steps. (b) Discrepancies between TENG-Heun and FEM solutions across multiple spatial resolutions (c) the scaling trend of discrepancies. between TENG-Heun and FEM solutions at $T = 4$. (d) Pairwise $L^2$ discrepancies among TENG-Heun, FEM-256, and spectral-1024 solvers.
  • Figure 5: Comparison between the exact solution and the TENG-BC prediction for the heat equation with dirichlet boundary at representative time instances. Top: exact solution. Middle: solution obtained by TENG-BC. Bottom: absolute error.
  • ...and 9 more figures