Multifidelity PhysicsConstrained Neural Networks With Minimax Architecture

Abstract

Data sparsity is still the main challenge to apply machine learning models to solve complex scientific and engineering problems. The root cause is the “curse of dimensionality” in training these models. Training algorithms need to explore and exploit in a very highdimensional parameter space to search the optimal parameters for complex models. In this study, a new scheme of multifidelity physicsconstrained neural networks with minimax architecture is proposed to improve the data efficiency of training neural networks by incorporating physical knowledge as constraints and sampling data with various fidelities. In this new framework, fully connected neural networks with two levels of fidelities are combined to improve the prediction accuracy. The lowfidelity neural network is used to approximate the lowfidelity data, whereas the highfidelity neural network is adopted to approximate the correlation function between the lowfidelity and highfidelity data. To systematically search the optimal weights of various losses for reducing the training time, the DualDimer algorithm is adopted to search highorder saddle points of the minimax optimization problem. The proposed framework is demonstrated with twodimensional heat transfer, phase transition, and dendritic growth problems, which are fundamental in materials modeling. With the same set of training data, the prediction error of the multifidelity physicsconstrained neural network with minimax architecture can be two orders of magnitude lower than that of the multifidelity neural network with minimax architecture.