Physics-Informed Regularization of Deep Neural Networks

This paper presents a novel physics-informed regularization method for training of deep neural networks (DNNs). In particular, we focus on the DNN representation for the response of a physical or biological system, for which a set of governing laws are known. These laws often appear in the form of differential equations, derived from first principles, empirically-validated laws, and/or domain expertise. We propose a DNN training approach that utilizes these known differential equations in addition to the measurement data, by introducing a penalty term to the training loss function to penalize divergence form the governing laws. Through three numerical examples, we will show that the proposed regularization produces surrogates that are physically interpretable with smaller generalization errors, when compared to other common regularization methods. Introduction Many science and engineering problems require repetitive simulation runs of a model with different input values. Examples of these problems include design optimization, model calibration, sensitivity analysis, what-if analysis, and design space exploration problems. However, in many real-world problems, obtaining a reliable outcome requires large number of these solves (typically for a partial differential equation), which can be prohibitive given the available resources. One way to alleviate this burden is to construct surrogate models [1] that mimic the solution or response surface. One example is building an analytical polynomial function for the displacement of a 2D plate at different locations.