This seminar is organized by the Scientific Computing group of CWI Amsterdam. The focus is on the application of Machine Learning (ML) and Uncertainty Quantification in scientific computing. Topics of interest include, among others:

- combination of data-driven models and (multi scale) simulations
- new ML architectures suited for scientific computing or UQ,
- incorporation of (physical) constraints into data-driven models,
- efficient (online) learning strategies,
- using ML for dimension reduction / creating surrogates,
- inverse problems using ML surrogates,

and any other topic in which some form of ML and/or UQ is used to enhance (existing) scientific computing methodologies. All applications are welcome, be it financial, physical, biological or otherwise.

For more information, or if you'd like to attend one of the talks, please contact Wouter Edeling of the SC group.

## Schedule upcoming talks

**6 September 2024 11h00 CET: Michael Abdelmalik (Eindhoven University of Technology): Neural Green's Operators for Parametric Partial Differential Equations**

We consider operator networks as promising machine learning tools for reduced order modeling of a wide range of physical systems described by partial differential equations (PDEs). This work introduces neural Green's operators (NGOs), a novel neural operator network architecture that learns the solution operator for a parametric family of linear partial differential equations (PDEs). Our construction of NGOs is derived directly from the Green's formulation of such a solution operator. Such a NGO acts as a surrogate for the PDE solution operator: it maps the PDE’s input functions (e.g. forcing, boundary conditions, PDE coefficients) to the solution. We apply NGOs to relevant canonical PDEs to demonstrate their efficacy and robustness as compared to a standard Deep Operator Networks [1], Variationally Mimetic Operator Networks [2] and Fourier Neural Operators [3]. Furthermore, we show that the explicit representation of the Green's function that is returned by NGOs enables the construction of effective preconditioners for numerical solvers for PDEs.

References

[1] Lu, Lu, et al. "Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.” In: Nat. Mach. Intell. 3.3 (2021).

[2] Dhruv Patel et al. “Variationally mimetic operator networks”. In: Comput. Methods Appl. Mech. Eng. 419 (2024).

[3] Zongyi Li et al. “Fourier neural operator for parametric partial differential equations”. In: arXiv preprint arXiv:2010.08895 (2020).

**12 September 2024 11h00 CET: Dimitrios Loukrezis (Siemens / TU Darmstadt): Scientific Machine Learning and Uncertainty Quantification for Predictive Digital Twins**

The Digital Twin (DT) is a cornerstone technology concept within the ongoing 4th industrial revolution. By connecting physical assets to their digital replicas, DTs offer unprecedented opportunities for innovation in industrial design and operation, for example, with respect to design optimization, decision support, and real-time monitoring. State of the art computational methods utilized within DT applications combine approaches stemming from traditional, physics-based computational science and engineering (CSE), with data-driven, machine learning (ML)-based techniques. This combination, often referred to as scientific machine learning (SciML), is widely considered to be the next step in the evolution of CSE and receives much attention within both industry and academia. At the same time, the models employed in a DT, physics-based, data-driven, or hybrid, will to some extent deviate from the corresponding real-world systems, commonly due to aleatory or epistemic uncertainties (or a mix thereof). In this context, uncertainty quantification (UQ) plays a crucial role in accompanying the DT's predictions with uncertainty metrics, thus providing robustness and reliability estimates. This talk will (i) present a general overview of DTs; (ii) present specific use-cases of SciML and UQ for DT applications; (iii) discuss currently missing aspects for realizing holistic DTs, i.e., data-driven, physics-conforming, and uncertainty-aware.