无需训练即可修正神经PDE求解器的物理约束偏差
Enforcing governing equation constraints in neural PDE solvers via training-free projections
- 用非线性优化和局部线性化方法对解进行后处理投影
- 在典型PDE上使约束违反减少，精度优于传统物理约束方法
- 适合需要高保真物理一致性的科学模拟场景
神经PDE求解器在科学模拟中常违背控制方程约束。虽然线性约束可低成本投影，但多数约束为非线性，使可行集投影复杂化。动力学PDE尤其困难，因约束引入长时间程依赖。本文评估两种无需训练的后处理投影方法：基于非线性优化的投影，以及利用雅克比-向量积和向量-雅克比积的局部线性化投影。通过分析代表性PDE的约束表现，发现两种方法均显著降低约束违反，提升精度，优于物理信息基线。
原文摘要 · Abstract (English)
Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long-range dependencies in time. In this work, we evaluate two training-free, post hoc projections of approximate solutions: a nonlinear optimization-based projection, and a local linearization-based projection using Jacobian-vector and vector-Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.
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