Learning governing equations from data is a central problem in scientific machine learning. Given noisy and incomplete observations of a physical process, the goal is to recover the underlying law. This is often approached by fitting a neural network or a dictionary of candidate terms to the observed dynamics. A good fit, however, does not determine identifiability of the law, its approximability by what is computed, physical consistency, or interpretability as a formula. These depend largely on how the learning problem is posed.

This talk formulates the learning task as a regularized inverse problem in function space and studies how these properties can be established. Over a parameterized class of candidate laws, one minimizes a model residual, a data misfit, and a regularizer, with the class and the regularizer as the design choices. With a sufficiently expressive class and suitable regularization, one obtains a regularization-based notion of identifiability, under which the regularization-minimizing law consistent with the data is unique. As the approximation scale grows and the regularization parameters are chosen accordingly, minimizers of the parameterized problem converge to that law. For classes carrying the structural constraints of the underlying physical model, the learned models are physically consistent and well-posed by construction at every finite approximation scale. For symbolic networks built from rational building blocks, the recovered law is a readable formula. The emphasis of this talk is analytical, and first numerical experiments illustrate the recovery in practice. 

Overall, the results show that identifiability, consistency, and interpretability are not competing objectives, but can be unified in one analysis-driven framework.

TBA

TBA