Artificial intelligence and machine learning are increasingly shaping how engineers analyze, design, and assess structures. In structural engineering, these tools enable optimization, rapid surrogate modeling, and decision-making under complex loading and environmental conditions. As simulations become more automated and data-driven, ensuring their reliability and robustness becomes increasingly important.

 

This talk discusses how computational mathematics supports trustworthy structural simulations in the AI era. Key topics include verification and validation, uncertainty quantification, and data quality, framed within the broader context of simulation governance. The focus is on how these ideas influence practical modeling choices and interpretation of results in engineering applications.

 

The presentation will highlight examples from advanced finite element analysis and AI-enabled structural modeling, including applications to structural optimization and reliability assessment under extreme snow loads. These examples illustrate how physics-based methods and data-driven tools can be combined effectively, while emphasizing the importance of uncertainty awareness when simulations inform safety-critical decisions.

About twenty years ago, Peres and Weiss generalised the classical Poisson limit theorem for appearances of words of increasing length in a sequence x. They showed that the theorem holds for almost every x with respect to the infinite uniform product measure. A natural question is whether this Poisson behaviour persists when the sequence is sampled according to a different product measure.

 

In our first result, we consider non-stationary product measures and show that there exists a quantitative threshold above which the Poisson limit theorem holds for almost every x, while below this threshold it may fail. In contrast, our second result shows that for a biased infinite product measure (a non-fair coin) the limiting behaviour is almost surely non-Poisson. This shows that the Poisson regime is specific to the equiprobable case and to small deviations from it.

 

This talk is based on works with Mike Hochman and Jon V. Kogan.

 

In this talk we will approach the following quantization problem: Assume that the same physical system is described classically by a Hamiltonian h_0 and quantumly by a Hamiltonian $H_0$. Is it possible to find a quantization procedure mapping classical constants of motion (or conserved quantities) of h_0 into quantum constant of motion of H_0?

 

We will show that the answer is positive for two basic Hamitonians: the Harmonic Oscillator and the (flat) Laplacian. The required quantization will be the canonical Weyl quantization. We will discuss the consequences of these results and the techniques applied to prove them as well. We will provide some explicit examples of families of constants of motion in both cases.

This presentation examines the use of Physics-Informed Neural Networks (PINNs), Variational Physics-Informed Neural Networks (VPINNs), Deep Ritz methods, and First Order System Least Squares (FOSLS) combined with stochastic quadrature rules, to solve parametric partial differential equations (PDEs). It begins by introducing parametric PDEs and how these neural network techniques can be used to solve them. The presentation then delves into the challenges of solving these PDEs, including optimization, regularity, and integration. It points out that while PINNs using strong formulations may have trouble with singular solutions, they handle integration better than weak formulation methods like VPINNs or Deep Ritz. To address these integration challenges, we propose the use of unbiased high-order stochastic quadrature rules for better integration and Regularity Conforming Neural Networks to deal with complex solutions and singularities.

 

Finally, the presentation discusses the broader significance of this research for solving parametric PDE problems and suggests directions for future research, and how FOSLS and PINNs may work better than VPINNs and Deep Ritz in different cases.