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Physics constrained learning for data-driven inverse modeling from sparse observations
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DOI：10.1016/j.jcp.2021.110938.png)
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Abstract
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Deep neural networks (DNN) can model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that measures the discrepancy between predictions and observations in some chosen norm. This loss function often includes the PDE constraints as a penalty term when only sparse observations are available. As a result, the PDE is only satisfied approximately by the solution. However, the penalty term typically slows down the convergence of the optimizer for stiff problems. We present a new approach that trains the embedded DNNs while numerically satisfying the PDE constraints. We developed an algorithm that enables differentiating both explicit and implicit numerical solvers in reverse-mode automatic differentiation. Our method allows the gradients of the DNNs and the PDE solvers to be computed in a unified framework. We demonstrate that our approach enjoys faster convergence and better stability in relatively stiff problems compared to the penalty method. Our approach could solve and accelerate a wide range of data-driven inverse modeling, where the physical constraints are described by PDEs and need to be satisfied accurately. (C) 2022 Elsevier Inc. All rights reserved.
Keywords:
Machine learning
Deep neural networks
Inverse problems
Numerical partial differential equations
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