### Mathematics Colloquium

** Marta D'Elia
**

Sandia National Lab

**Title:** Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws.

**Date:** Friday, March 26, 2021

**Place and Time:** Zoom, 3:05-3:55 pm

**Abstract.**
Nonlocal models are characterized by integral operators that embed lengthscales in their definition. As such, they are preferable to classical partial differential equation models in situations where the dynamics of a system is affected by the small scale behavior, yet the small scales would require prohibitive computational cost to be treated explicitly. In this sense, nonlocal models can be considered as coarse-grained, homogenized models that, without resolving the small scales, are still able to accurately capture the system's global behavior. However, nonlocal models depend on "kernel functions", the integrands, that are often defined ad hoc or hand tuned; this fact hinders the usability of these model.
We propose to learn optimal kernel functions from high fidelity data by combining machine learning algorithms, known physics, and nonlocal theory. This combination guarantees that the resulting model is mathematically well-posed and physically consistent. Furthermore, by learning the operator rather than a surrogate for the solution, these models generalize well to settings that are different from the ones used during training. We apply this "operator regression" technique to wave propagation in heterogeneous materials and to molecular dynamics. In both cases, the machine-learned nonlocal operator embeds material properties in the kernel function and allows for accurate predictions at much coarser scales than micro or molecular scales.