PINNACLE: PINN Adaptive ColLocation and Experimental points selection

TL;DR

PINNACLE is a novel algorithm that adaptively selects and optimizes all training point types in Physics-Informed Neural Networks, improving training efficiency and accuracy by considering interactions among points.

Contribution

It introduces the first joint optimization method for collocation and experimental point selection in PINNs, based on NTK analysis and adaptive adjustment during training.

Findings

PINNACLE outperforms existing point selection methods in various PINN tasks.

The method adaptively balances point types, leading to better training dynamics.

Theoretical analysis links PINNACLE's criterion to generalization error.

Abstract

Physics-Informed Neural Networks (PINNs), which incorporate PDEs as soft constraints, train with a composite loss function that contains multiple training point types: different types of collocation points chosen during training to enforce each PDE and initial/boundary conditions, and experimental points which are usually costly to obtain via experiments or simulations. Training PINNs using this loss function is challenging as it typically requires selecting large numbers of points of different types, each with different training dynamics. Unlike past works that focused on the selection of either collocation or experimental points, this work introduces PINN Adaptive ColLocation and Experimental points selection (PINNACLE), the first algorithm that jointly optimizes the selection of all training point types, while automatically adjusting the proportion of collocation point types as training progresses. PINNACLE uses information on the interaction among training point types, which had not been considered before, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). We theoretically show that the criterion used by PINNACLE is related to the PINN generalization error, and empirically demonstrate that PINNACLE is able to outperform existing point selection methods for forward, inverse, and transfer learning problems.