Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanics

TL;DR

This paper introduces a Sobolev neural network with residual weighting that incorporates sensitivity information to improve surrogate modeling in linear and nonlinear mechanics, enhancing training efficiency and accuracy.

Contribution

It proposes a novel Sobolev training approach with residual weighting and adaptive optimization, improving neural network surrogate performance in computational mechanics.

Findings

Enhanced training convergence with sensitivity data inclusion

Residual weighting improves model accuracy and precision

Adaptive residual weight optimization benefits surrogate modeling

Abstract

Areas of computational mechanics such as uncertainty quantification and optimization usually involve repeated evaluation of numerical models that represent the behavior of engineering systems. In the case of complex nonlinear systems however, these models tend to be expensive to evaluate, making surrogate models quite valuable. Artificial neural networks approximate systems very well by taking advantage of the inherent information of its given training data. In this context, this paper investigates the improvement of the training process by including sensitivity information, which are partial derivatives w.r.t. inputs, as outlined by Sobolev training. In computational mechanics, sensitivities can be applied to neural networks by expanding the training loss function with additional loss terms, thereby improving training convergence resulting in lower generalisation error. This improvement is shown in two examples of linear and non-linear material behavior. More specifically, the Sobolev designed loss function is expanded with residual weights adjusting the effect of each loss on the training step. Residual weighting is the given scaling to the different training data, which in this case are response and sensitivities. These residual weights are optimized by an adaptive scheme, whereby varying objective functions are explored, with some showing improvements in accuracy and precision of the general training convergence.