title: WHAT TO DO IF SPARSE REPRESENTATION LEARNING FAILS UNEXPECTEDLY?
abstract: Learning physical equations from data is essential for scientific discovery and engineering modeling. However, most of the existing methods rely on two rules: (1) learn a sparse representation to fit data and (2) check if the loss objective function satisfies error thresholds. This paper illustrates that such conditions are far from sufficient. Specifically, we show that sparse non-physical approximations exist with excellent fitting accuracy, but fail to adequately model the situation. To fundamentally resolve the data-fitting problem, we propose a physical neural network (PNN) utilizing “Range, Inertia, Symmetry, and Extrapolation” (RISE) constraints. RISE is based on a complete analysis for the generalizability of data properties for physical systems. The first three techniques focus on the definition of physics in space and time. The last technique of extrapolation is novel based on active learning without an inquiry, using cross-model validation. We validate the proposed PNN-RISE method via a synthetic dataset, power system dataset, and mass-damper system dataset. Numerical results show the universal capability of the PNN-RISE approach to quickly identify the hidden physical models without local optima, opening the door for the fast and highly accurate discovery of the physical laws or systems with external loads.
authors: Jingyi Yuan Haoran Li Erik Blasch Yang Weng
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