用自然梯度统一处理各类边界条件，提升神经PDE求解精度与稳定性。
TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions
- 基于时间演化自然梯度，每步优化同时满足内部动力学和边界条件。
- 在扩散、传输、非线性PDE上实现高精度求解，优于传统PINN方法。
- 无需调参即可稳定演进，适合复杂边界条件的科学计算场景。
由于长时间误差累积及通用边界条件难以施加，基于神经网络求解时变偏微分方程仍具挑战。本文提出TENG-BC，一种基于时间演化自然梯度的高精度神经PDE求解器，可有效处理一般边界约束。在每个时间步，TENG-BC执行感知边界的优化，联合施加内部动力学与各类边界条件（包括Dirichlet、Neumann、Robin及混合型），并可在统一框架下实现自然梯度解释，从而避免繁琐的惩罚参数调优，确保稳定时间演化。在扩散、输运及非线性PDE等多类基准测试中，TENG-BC在相近采样预算下达到求解器级精度，显著优于传统求解器与物理信息神经网络（PINN）基线。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。