Physics-informed learning under mixing: How physical knowledge speeds up learning
This addresses a major problem in physics-informed machine learning for researchers and practitioners dealing with dependent data, offering a theoretical advancement that is incremental but specific to this bottleneck.
The paper tackles the challenge of how physical prior knowledge affects learning rates with dependent data, proving that aligned physical information improves the learning rate from the slow Sobolev minimax rate to the fast optimal i.i.d. rate without sample-size deflation.
A major challenge in physics-informed machine learning is to understand how the incorporation of prior domain knowledge affects learning rates when data are dependent. Focusing on empirical risk minimization with physics-informed regularization, we derive complexity-dependent bounds on the excess risk in probability and in expectation. We prove that, when the physical prior information is aligned, the learning rate improves from the (slow) Sobolev minimax rate to the (fast) optimal i.i.d. one without any sample-size deflation due to data dependence.