TINNs让神经网络随时间动态调整，更准更快求解时变偏微分方程。
TINNs: Time-Induced Neural Networks for Solving Time-Dependent PDEs
- 将网络权重设计为随时间变化的函数，实现时空特征解耦
- 相对误差降低至PINN的1/4，收敛速度提升10倍
- 适合需高精度动态建模的物理仿真场景
物理信息神经网络（PINNs）通过学习无网格、可微的解来求解时变偏微分方程（PDE），可在任意时空点评估。但标准的时空PINNs将时间作为输入，却在所有时刻共享同一组网络权重，迫使相同特征表示显著不同的动态行为，导致误差性能下降，并在联合施加PDE、边界和初值约束时可能引发训练不稳定。本文提出时间诱导神经网络（TINNs），将网络权重参数化为时间的可学习函数，使有效空间表征随时间演化，同时保持共享结构。该方法自然形成非线性最小二乘问题，采用Levenberg-Marquardt算法高效优化。在多种时变PDE上的实验表明，相较PINNs及强基线，相对误差最高降低4倍，收敛速度最快提升10倍。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) solve time-dependent partial differential equations (PDEs) by learning a mesh-free, differentiable solution that can be evaluated anywhere in space and time. However, standard space-time PINNs take time as an input but reuse a single network with shared weights across all times, forcing the same features to represent markedly different dynamics. This coupling degrades error performance and can destabilize training when enforcing PDE, boundary, and initial constraints jointly. We propose Time-Induced Neural Networks (TINNs), a novel architecture that parameterizes the network weights as a learned function of time, allowing the effective spatial representation to evolve over time while maintaining shared structure. The resulting formulation naturally yields a nonlinear least-squares problem, which we optimize efficiently using a Levenberg-Marquardt method. Experiments on various time-dependent PDEs show up to 4 times improved relative error and 10 times faster convergence compared to PINNs and strong baselines.
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