From deterministic physics-based to probabilistic data-driven modeling: Diffusion-based prediction of strain fields in deep drawing processes

Abstract

A new perspective is adopted to solve boundary value problems in structural mechanics. In a compact form, they are described as the mapping from an input vector that defines a mechanical system to an output image that describes mechanical fields. This mapping is then directly learned from data using a deterministic transpose convolutional neural network (CNN) model. Here, we apply this approach to predict the strain fields in deep drawing. The model-specifying input variables include the material properties, the forming tool geometries and the punch displacement. Training data comprising 10,000 pairs of input vectors and output images is generated through finite element (FE) simulations. It is shown that the trained CNN is able to make reliable predictions, including complex deformation patterns associated with wrinkling. To facilitate the training on real experimental data, we also develop a diffusion denoising probabilistic (DDP) model. Different from the CNN, the DDP model learns an output image-generating distribution from data sets with missing input information. While the DDP is able to perform the same tasks (with comparable accuracy) as the deterministic CNN, it also provides meaningful probabilistic predictions when an input variable such as the friction coefficient is unknown. The successful adoption of the probabilistic neural network approach is seen as an important step towards the development of data-driven models that exceed the predictive capabilities of traditional models. This approach is expected to become particularly valuable in applications where system-defining variables are not measurable or the physical understanding is incomplete.