Abstract: Accurate and efficient simulation of physical phenomena governed by Partial Differential Equations (PDEs) is a major computational challenge, most of all when related to time-consuming activities such as parameter estimation, control and sensitivity analysis. We deal specifically with real-time problems defined by varying boundary conditions, where a parametric instance modifies not only the physics of the PDE but also the computational domain and the application of boundary constraints.

Current solution methods based on classical Model Order Reduction (MOR) techniques encounter a fundamental bottleneck in this scenario. Parametric boundaries necessitate, in most of the cases, re-meshing and complete re-formulation of the discrete problem for each new configuration, making the use of standard MOR techniques impossible for real-time design and optimization.

We propose a novel methodology based on Graph-Instructed Neural Networks (GINNs) to overcome this limitation. The GINN framework effectively learns the mapping between the parametric description of the computational domain and the resulting PDE solution. The proposed GINN-based model represents highly complex parametric PDEs efficiently, serving as a valuable asset for real-case and many-query complex problems.

Biography: Sandra Pieraccini is a Full Professor of Numerical Analysis at Politecnico di Torino. She obtained her Master’s degree in Mathematics and a Ph.D. in Applied Mathematics and Operations Research. Her main research interests include Numerical Optimization and Uncertainty Quantification, which she has recently integrated with investigations in the field of deep learning. She is the author of approximately 60 scientific publications in international journals and conference proceedings. She currently serves as the Deputy Head of the Mathematical Science Department “G. L. Lagrange” at Politecnico di Torino, and as the Head of the working group on “Mathematics for Artificial Intelligence and Machine Learning” within the Unione Matematica Italiana (Italian Mathematical Union)

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