Data Driven Surrogate Frameworks for Computational Mechanics: Bayesian and Geometric Deep Learning Approaches

In modern applications, high-fidelity computational models are often impractical due to their slow performance and also lack information about the certainty of their predictions. Deep learning techniques have recently emerged as a powerful tool for accelerating such predictions. However, these techniques can be inefficient when confronted with larger and more complex problems. This thesis introduces innovative deep learning surrogate frameworks that are scalable, robust, require minimum hyper-parameter tuning, are fast at the inference stage, and are accurate in forecasting non-linear deformation responses of solid objects. These surrogate frameworks are constructed using various deep learning techniques under deterministic as well as Bayesian settings. Bayesian frameworks enable us to capture uncertainties and provide a means to trust the predictions of the neural network approaches.
This thesis introduces a new geometric deep learning framework, called MAgNET (Multi-channel Aggregation Network). MAgNET is designed to handle large-dimensional graph-structured data using an encoder-decoder architecture. MAgNET is built upon the novel MAg (Multichannel Aggregation) operation, which generalises the concept of multi-channel local operations found in convolutional neural networks to arbitrary non-grid inputs. The MAg layers are combined with the novel graph pooling/unpooling operations to form a powerful graph U-Net architecture capable of efficiently performing supervised learning on large-dimensional graph-structured data, like complex meshes. Additionally, the thesis demonstrates the use of state-of-the-art
attention-based networks, which have revolutionized various engineering fields but have remained unexplored for their uses in the field of computational mechanics.
We demonstrate the efficiency and versatility of the proposed frameworks by applying them to surrogate modeling for non-linear finite element simulations. Our suggested methods, particularly the MAgNET architecture, possess broad applicability, enabling researchers and practitioners to explore novel modeling scenarios and applications. Through the open sharing of the source codes and datasets employed, this thesis not only makes a significant contribution to the field of surrogate modeling in mechanics but also paves the way for numerous research opportunities for their utilisation in various engineering and scientific applications.