PINNACLE: PINN ADAPTIVE COLLOCATION AND EXPERIMENTAL POINTS SELECTION

PINNACLE: PINN ADAPTIVE COLLOCATION AND EXPERIMENTAL POINTS SELECTION

2024 | Gregory Kang Ruey Lau, Apivich Hemachandra, See-Kiong Ng & Bryan Kian Hsiang Low
PINNACLE is a novel algorithm that jointly optimizes the selection of all training point types for Physics-Informed Neural Networks (PINNs), including collocation points and experimental points. Unlike previous methods that focused on selecting either collocation or experimental points separately, PINNACLE automatically adjusts the proportion of each type as training progresses. It leverages information on the interaction among training point types, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). The algorithm uses this information to define a new convergence criterion, which is theoretically related to the PINN generalization error. Empirically, PINNACLE outperforms existing point selection methods for forward, inverse, and transfer learning problems. The algorithm is based on the NTK spectrum and the convergence degree criterion, which is computed using Nystrom approximation. PINNACLE includes two variants: one that maximizes the convergence degree and another that considers the evolution of the empirical NTK (eNTK). The algorithm is evaluated on various PDEs and shows improved performance in forward, inverse, and transfer learning tasks. The results demonstrate that PINNACLE can efficiently select training points, leading to faster convergence and better generalization.PINNACLE is a novel algorithm that jointly optimizes the selection of all training point types for Physics-Informed Neural Networks (PINNs), including collocation points and experimental points. Unlike previous methods that focused on selecting either collocation or experimental points separately, PINNACLE automatically adjusts the proportion of each type as training progresses. It leverages information on the interaction among training point types, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). The algorithm uses this information to define a new convergence criterion, which is theoretically related to the PINN generalization error. Empirically, PINNACLE outperforms existing point selection methods for forward, inverse, and transfer learning problems. The algorithm is based on the NTK spectrum and the convergence degree criterion, which is computed using Nystrom approximation. PINNACLE includes two variants: one that maximizes the convergence degree and another that considers the evolution of the empirical NTK (eNTK). The algorithm is evaluated on various PDEs and shows improved performance in forward, inverse, and transfer learning tasks. The results demonstrate that PINNACLE can efficiently select training points, leading to faster convergence and better generalization.
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[slides] PINNACLE%3A PINN Adaptive ColLocation and Experimental points selection | StudySpace