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When: Thursday, October 10^th, 2024 from 2:30 to 3:45 p.m. EDT

Speaker: Assad A. Oberai (University of Southern California)

Abstract: We present a novel probabilistic approach for generating multi-fidelity data while accounting for errors inherent in both low- and high-fidelity data. In this approach a graph Laplacian constructed from the low-fidelity data is used to define a multivariate Gaussian prior density for the coordinates of the true data points. In addition, a few high-fidelity data points are used to construct a conjugate likelihood term. Thereafter, Bayes rule is applied to derive an explicit expression for the posterior density which is also multivariate Gaussian. The maximum a posteriori (MAP) estimate of this density is selected to be the optimal multi-fidelity estimate. It is shown that the MAP estimate and the covariance of the posterior density can be determined through the solution of a linear system of equations. Thereafter, two methods, one based on spectral truncation, and another based on a low-rank approximation, are developed to solve these equations efficiently. The multi-fidelity approach is tested on a variety of problems in solid and fluid mechanics with data that represents vectors of quantities of interest and discretized spatial fields in one and two dimensions. The results demonstrate that by utilizing a small fraction of high-fidelity data, the multi-fidelity approach can significantly improve the accuracy of a large collection of low-fidelity data points.

This is joint work with Orazio Pinti from USC and Jeremy M. Budd and Franca Hoffmann from Caltech

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When: Wednesday, September 18^th, 2024 from 2:30 to 3:30 p.m. EDT

Speaker:  Justin Dong (Lawrence Livermore National Laboratory)

Abstract: Recently, neural networks have been utilized for tasks that have traditionally been in the domain of scientific computing, for instance the forward approximation problem for partial differential equations (PDEs). While neural networks are known to satisfy a universal approximation property, they are difficult to train and often stagnate prematurely. In particular, neural networks often fail to deliver an approximation with controllable error – increasing the number of parameters in the network does not improve approximation error beyond a certain point.

We present some recent developments towards neural network-based numerical methods that provide error control. In the first part of this talk, we introduce the Galerkin neural network framework which constructs a finite-dimensional subspace whose basis functions are the realizations of a sequence of neural networks. The hallmark of this framework is an a-posteriori error estimator for the energy error that provides the user with full control of the approximation error. In the second part of this talk, we discuss issues of well-posedness as it pertains to loss functions used to train neural networks. Most common loss functions proposed in the literature for physics-informed learning may be viewed as the functionals to corresponding least squares variational problems. Viewed in this light, we demonstrate that many such loss functions lead to ill-posed variational problems and present recent work towards constructing well-posed loss functions for arbitrary boundary value problems.