This paper proposes a sparse regression strategy for discovery of ordinary and partial differential equations from incomplete and noisy data. Inference is performed over both equation parameters and state variables using a statistically motivated likelihood function. Sparsity is enforced by a selection algorithm which iteratively removes terms and compares models using statistical information criteria. Large scale optimization is performed using a second-order variant of the Levenberg-Marquardt method, where the gradient and Hessian are computed via automatic differentiation. Illustrations involving canonical systems of ordinary and partial differential equations are used to demonstrate the flexibility and robustness of the approach. Accurate reconstruction of systems is found to be possible even in extreme cases of limited data and large observation noise.

翻译：本文提出一种稀疏回归策略，用于从不完整且含噪声的数据中发现常微分方程与偏微分方程。通过基于统计动机的似然函数，对方程参数和状态变量同时进行推断。稀疏性通过选择算法强制实施，该算法迭代移除方程项并利用统计信息准则比较模型。大规模优化采用Levenberg-Marquardt方法的二阶变体实现，其梯度与Hessian矩阵通过自动微分计算。通过典型常微分方程与偏微分方程系统的示例，验证了该方法具有灵活性与鲁棒性。研究发现即使在数据极度有限且观测噪声较大的极端情况下，仍能实现系统的精确重构。