Physics-informed kernel learning
This work addresses the challenge of efficiently incorporating physical constraints into machine learning models for applications like hybrid modeling and PDE solving, representing an incremental improvement over existing methods.
The paper tackles the problem of integrating physical priors into machine learning by proposing physics-informed kernel learning (PIKL), a kernel regression approach with Fourier approximations, and demonstrates that it outperforms physics-informed neural networks in accuracy and computation time, and surpasses traditional PDE solvers in noisy boundary conditions.
Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data-driven term and a partial differential equation (PDE) regularization. Building on the formulation of the problem as a kernel regression task, we use Fourier methods to approximate the associated kernel, and propose a tractable estimator that minimizes the physics-informed risk function. We refer to this approach as physics-informed kernel learning (PIKL). This framework provides theoretical guarantees, enabling the quantification of the physical prior's impact on convergence speed. We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time. Additionally, we identify cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions.