Graduate Students Seminar

Speaker 1: Weixin Wang

Title

Attentive Neural Processes

Abstract

Neural Processes (NPs) (Garnelo et al., 2018a;b) approach regression by learning to map a context set of observed input-output pairs to a distribution over regression functions. In comparison with deep neural networks, NPs outputs a distribution (a stochastic process) as well as the uncertainty over functions rather than a single deterministic function. However, NPs suffer a fundamental drawback of underfitting, giving inaccurate predictions at the inputs of the observed data they condition on. Attentive Neural Processes((A)NP) address this issue by employing an attention mechanism into NPs. (A)NP utilizes self-attention and cross attention in its encoder to better summarize the context set. We show that this greatly improves the accuracy of predictions.

Speaker 2: Ellie Gurvich

Title

Strong and Weak Solutions for a Biot-Stokes FPSI via a Semigroup Approach

Abstract

A filtration system, comprising a Biot poroelastic solid coupled to an incompressible Stokes free-flow, is considered in 3D. Across the 2D interface, the Beavers-Joseph-Saffman coupling conditions are enforced. A semigroup approach circumvents typical issues associated with mismatched trace regularities at the interface. The linear hyperbolic-parabolic coupled problem in the fully inertial and non-degenerate case is posed through a dynamics operator on an appropriate energy space. Strong and generalized solutions are obtained via $C_0$-semigroup generation for the dynamics operator. A standard argument by density is shown to yield weak solutions, including the degenerate cases where the Biot compressibility of the constituents vanishes. Thus, for the inertial Biot-Stokes filtration, we provide a clear elucidation of strong and weak solutions and their regularity with associated energy estimates.