Machine Learning based solutions of PDE with applications in engineering and mechanics

Prof Timon Rabczuk

Bauhaus University Weimar

Abstract: This presentation highlights recent advancements in Scientific Machine Learning (SciML) for modeling physical systems governed by partial differential equations (PDEs). It will first compare and present an overview of SciML approaches including Physics-Informed Neural Networks (PINNs), Deep Energy Methods (DEMs) and Neural Operators for analyzing mechanical problems. Then, two advanced neural operator frameworks will be introduced: one for static problems and one for dynamic problems. For static PDEs, the Variational Physics-Informed Neural Operator (VINO) combines the generalization power of neural operators with the accuracy and stability of energy-based formulations. VINO minimizes the variational form of PDEs rather than pointwise residuals, enabling training without labeled data and significantly improving performance over existing machine learning methods, particularly as mesh resolution increases. Its element-based discretization enhances scalability and physical fidelity, addressing key limitations in existing physics-informed models. For dynamic problems, we present the so-called Step Aware Neural Operator (SANO), designed for efficient multi-step predictions in time-dependent PDEs. SANO incorporates time-step-specific projections and message-passing mechanisms, capturing long-term dependencies while avoiding error accumulation, and demonstrates strong performance across a range of phase field models. Finally, a hybrid approach will be proposed that uses the output of neural operators as an initial guess for iterative solvers, significantly improving the computational time for challenging problems by combining data-driven prediction with the robustness of numerical methods.

Bio: Timon has graduated at University of Karlsruhe in January 2002 and spend almost 4 years as postdoctoral fellow at the Computational Mechanics group of Prof. Ted Belytschko in Northwestern University before joining the University of Munich. In 2007, he was appointed Senior Lecturer in the Mechanical Engineering Department at University of Canterbury in New Zealand before recruited as chaired professor of Computational Mechanics at Bauhaus University Weimar in 2009. He has published more than 800 papers in international SCI journals and is on editorial boards of several journals. He is member of the European Academy of Sciences, Academia Europaea and European Academy of Sciences and Arts. His key research area is Computational Mechanics with particular focus on computational methods for the solution of PDEs with applications in Engineering, Mechanics and materials science.