用控制理论优化物理信息神经网络的训练，提升收敛精度与鲁棒性。
A Control Perspective on Training PINNs
- 将PINN训练视为带扰动的随机控制系统，动态调整物理损失权重。
- 积分控制器在模型准确时实现高精度稳定收敛，漏积分器更适应模型偏差。
- 为PINN提供理论保障，适合需要可靠训练的科学计算场景。
本文从控制理论视角研究物理信息神经网络（PINNs）的训练问题。通过带重采样的梯度下降法，将训练动态解释为渐近等价于带有过程扰动和测量噪声的随机控制仿射系统。在此框架下，提出两种动态调节物理损失权重的控制器：积分控制器与漏积分控制器。理论上分析了它们在精度-鲁棒性权衡下的渐近性质，并在简单例子上进行验证。数值结果表明，当物理模型正确时，积分控制器可实现精确且鲁棒的收敛；而在模型不匹配情况下，漏积分控制器表现更优。本工作为PINN的收敛性保证及针对性训练算法设计迈出第一步。
原文摘要 · Abstract (English)
We investigate the training of Physics-Informed Neural Networks (PINNs) from a control-theoretic perspective. Using gradient descent with resampling, we interpret the training dynamics as asymptotically equivalent to a stochastic control-affine system, where sampling effects act as process disturbances and measurement noise. Within this framework, we introduce two controllers for dynamically adapting the physics weight: an integral controller and a leaky integral controller. We theoretically analyze their asymptotic properties under the accuracy-robustness trade-off, and we evaluate them on a toy example. Numerical evidence suggests that the integral controller achieves accurate and robust convergence when the physical model is correct, whereas the leaky integrator provides improved performance in the presence of model mismatch. This work represents a first step toward convergence guarantees and principled training algorithms tailored to the distinct characteristics of PINN tasks.
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