Simon Tavener

Troy Butler, Don Estep, Simon Tavener et al.

We compute approximate solutions to inverse problems for determining parameters in differential equation models with stochastic data on output quantities. The formulation of the problem and modeling framework define a solution as a probability measure on the parameter domain for a given $σ-$algebra. In the case where the number of output quantities is less than the number of parameters, the inverse of the map from parameters to data defines a type of generalized contour map. The approximate contour maps define a geometric structure on events in the $σ-$algebra for the parameter domain. We develop and analyze an inherently non-intrusive method of sampling the parameter domain and events in the given $σ-$algebra to approximate the probability measure. We use results from stochastic geometry for point processes to prove convergence of a random sample based approximation method. We define a numerical $σ-$algebra on which we compute probabilities and derive computable estimates for the error in the probability measure. We present numerical results to illustrate the various sources of error for a model of fluid flow past a cylinder.

Quincy Hershey, Randy Paffenroth, Harsh Pathak et al.

Neural networks (NN) can be divided into two broad categories, recurrent and non-recurrent. Both types of neural networks are popular and extensively studied, but they are often treated as distinct families of machine learning algorithms. In this position paper, we argue that there is a closer relationship between these two types of neural networks than is normally appreciated. We show that many common neural network models, such as Recurrent Neural Networks (RNN), Multi-Layer Perceptrons (MLP), and even deep multi-layer transformers, can all be represented as iterative maps. The close relationship between RNNs and other types of NNs should not be surprising. In particular, RNNs are known to be Turing complete, and therefore capable of representing any computable function (such as any other types of NNs), but herein we argue that the relationship runs deeper and is more practical than this. For example, RNNs are often thought to be more difficult to train than other types of NNs, with RNNs being plagued by issues such as vanishing or exploding gradients. However, as we demonstrate in this paper, MLPs, RNNs, and many other NNs lie on a continuum, and this perspective leads to several insights that illuminate both theoretical and practical aspects of NNs.

Lorenz Berger, Rafel Bordas, David Kay et al.

A stabilized conforming mixed finite element method for the three-field (displacement, fluid flux and pressure) poroelasticity problem is developed and analyzed. We use the lowest possible approximation order, namely piecewise constant approximation for the pressure and piecewise linear continuous elements for the displacements and fluid flux. By applying a local pressure jump stabilization term to the mass conservation equation we ensure stability and avoid pressure oscillations. Importantly, the discretization leads to a symmetric linear system. For the fully discretized problem we prove existence and uniqueness, an energy estimate and an optimal a-priori error estimate, including an error estimate for the divergence of the fluid flux. Numerical experiments in 2D and 3D illustrate the convergence of the method, show the effectiveness of the method to overcome spurious pressure oscillations, and evaluate the added mass effect of the stabilization term.