用约束优化提升微分方程参数估计精度，噪声数据下仍稳定有效。
PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks
- 将训练转为带约束的微分优化，动态平衡数据与方程残差损失
- 在四个经典物理生物模型中实现噪声数据下的高精度参数反演
- 适用于前向求解昂贵的场景，适合科学建模与逆问题研究者
从测量数据中估计微分方程参数是定量科学中的常见反问题。物理信息神经网络（PINNs）在稀疏观测和系统信息不全时表现出色，但存在收敛困难、稳定性差、过拟合及损失函数设计复杂等问题。本文提出PINNverse，一种将学习过程重构为约束微分优化问题的训练范式。该方法在训练中动态平衡数据损失与微分方程残差损失，防止过拟合，并结合改进的乘子法，可实现对帕累托前沿任意点的收敛。我们在四个经典常微分方程与偏微分方程模型（涵盖物理与生物学）中验证了其在噪声数据下的鲁棒性与准确性。该方法尤其适用于前向问题求解成本高的情况，支持精确参数推断。
原文摘要 · Abstract (English)
Parameter estimation for differential equations from measured data is an inverse problem prevalent across quantitative sciences. Physics-Informed Neural Networks (PINNs) have emerged as effective tools for solving such problems, especially with sparse measurements and incomplete system information. However, PINNs face convergence issues, stability problems, overfitting, and complex loss function design. Here we introduce PINNverse, a training paradigm that addresses these limitations by reformulating the learning process as a constrained differential optimization problem. This approach achieves a dynamic balance between data loss and differential equation residual loss during training while preventing overfitting. PINNverse combines the advantages of PINNs with the Modified Differential Method of Multipliers to enable convergence on any point on the Pareto front. We demonstrate robust and accurate parameter estimation from noisy data in four classical ODE and PDE models from physics and biology. Our method enables accurate parameter inference also when the forward problem is expensive to solve.
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