Physics-informed machine learning combines data with structural constraints, often in the form of differential information. Kernel methods can be used for learning with linear differential constraints, yielding powerful models which avoid some of the pitfalls incurred with physics-informed neural networks, and remain theoretically tractable. I will introduce Physics-Informed KernelS (PIKS), a formalization of the physics-informed learning problem in reproducing kernel Hilbert spaces. In particular, I will discuss the asymptotic behaviour of PIKS in the misspecified setting, where minimal assumptions are required on the target function, which in particular does not need to belong to the hypothesis space. The presented results extend the learning guarantees of kernel methods to realistic settings in which the target smoothness is unknown: given a universal kernel (which include e.g. Gaussian and Matérn kernels), PIKS learns the target function in a way which is consistent with the physical constraints even in the misspecified setting. I will conclude by presenting numerical experiments, to illustrate the practical applications of the framework on tasks ranging from solving PDEs to general learning with differential data.
Joachim Bona-Pellissier completed a PhD in 2023 in applied mathematics in Toulouse, France, where he studied theoretical properties of neural networks, such as identifiability and implicit regularization. After that, he joined the MaLGa center in Genoa as a postdoctoral researcher. His current focus is on physics-informed machine learning and kernel methods.