An operator preconditioning perspective on training in physics-informed machine learning
This addresses a key bottleneck in training physics-informed models for researchers and practitioners in scientific computing, though it is incremental as it builds on existing operator theory.
The paper tackles the problem of slow or infeasible training in physics-informed machine learning methods like PINNs by linking it to the conditioning of a specific differential operator, showing that preconditioning this operator improves training efficiency.
In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.