Numerical Methods for Partial Differential Equations Seminar

One of the primary factors hindering the adoption of transformative low-emissions aircraft designs is the time-consuming (taking years) and costly (costing billions of dollars) experimental campaigns required during the design cycle. Computational fluid dynamics (CFD) might accelerate the process and alleviate the cost. However, current turbulence models do not meet the stringent accuracy requirements demanded by the industry. Here, we have devised a new closure model for CFD to bridge the gap between our current predictive capabilities and those required by the aerospace industry. This model, referred to as the building-block-flow model, conceives the flow as a collection of simple units that contain the essential flow physics necessary to predict complex flows. The approach is implemented using two artificial neural networks: a classifier that identifies the contribution of each building block in the flow, and a predictor that estimates the effect of missing scales through a combination of the building-block units. The training data are directly obtained from CFD with exact modeling for mean quantities to ensure consistency with the numerical discretization. The model's output is accompanied by confidence in the prediction, which is used for uncertainty quantification. The model is validated in realistic aircraft configurations.

We have deduced a weakly nonlinear, weakly dispersive Boussinesq system for water waves on a 1D branching channel, namely on a graph. The reduced model requires a compatibility condition at the graph’s node, where the main reach bifurcates into two reaches. Our new nonlinear compatibility condition arises from a stationary shock condition and generalizes that found in Stoker (1957), used since then. Our numerical method uses the Schwarz-Christoffel (conformal) mapping in order to generate a boundary fitted coordinate system for the forked channel region. This allows for general branching angles. We present numerical simulations comparing solitary waves on the 1D graph model with results of the (parent) 2D model, where a compatibility condition is not needed. It is shown that the condition given in Stoker is of limited accuracy.

Recovering sparse generative models from limited and noisy measurements presents a significant and complex challenge. Given that the available data is frequently inadequate and affected by noise, assessing the resulting uncertainty in the relevant parameters is crucial. Notably, this parameter uncertainty directly impacts the reliability of predictions and decision-making processes.

In this talk, we explore the sparsity-promoting hierarchical Bayesian learning framework, which facilitates the quantification of uncertainty in parameter estimates by treating involved quantities as random variables and leveraging the posterior distribution.

Within the Bayesian framework, sparsity promotion and computational efficiency can be attained with hierarchical models with conditionally Gaussian priors and gamma hyper-priors.

Parts of this talk are joint work with Anne Gelb (Dartmouth) and Youssef Marzouk (MIT).

Turbulence is a high-dimensional dynamical system with known equations of motion. It can be numerically integrated, but the simulation results are also high-dimensional and hard to interpret. Lower-dimensional models are not dynamical systems, because some dynamics is discarded in the projection, and a stochastic Perron-Frobenius operator substitutes the equations of motion. Using as example turbulent flows at moderate but non-trivial Reynolds number, we show that particularly deterministic projections can be identified by either Monte-Carlo or exhaustive testing, and can be interpreted as coherent structures. We also show that they can be used to construct data-driven 'fake' models that retain many of the statistical characteristics of the real flow.

The performance and safety of ocean systems such as underwater robots, aquaculture installations, and hydrofoils, are ultimately governed by non-linear flows and fluid-structure interactions. Performing high-fidelity simulations of such problems requires the ability to handle complex, moving domains and multiphysics phenomena. Immersed methods are widely used for this category of problems, because of their ability to handle complex domains with moving boundaries on structured grids without the need to generate body-fitted meshes and associated remeshing. A disadvantage of these methods is the complexity involved in extending their accuracy beyond first or second order, especially for quantities near or on the boundary or interface. Another challenge is their generalization to arbitrary boundary and interface conditions. Consequently, the vast majority of existing embedded boundary methods achieve first or second order accuracy in space.

In this talk I will detail our progress towards increasing the efficiency of adaptive-grid flow problems with immersed moving boundaries and interfaces. First, I will discuss our approach for high-order finite-difference based immersed boundary and interface discretization, and share our error and stability analyses. I will demonstrate the effectiveness of the approach in a variety of problems including a second-order 2D vorticity-velocity Navier-Stokes solver, high-order 2D linear elasticity problems, and high-order 3D advection-diffusion problems. Second, I will present our methodology for 3D multiresolution grid adaptation techniques using a wavelet-based analysis implemented within a scalable parallel software framework. This leads to a predictable convergence of the solution error with respect to the adaptation criteria, while the wavelet order effectively controls the compression ratio.

While we can simulate complex physical systems with millions of degrees of freedom, inferring these parameters from data is complicated due to their high dimensionality. It would be nice if we could focus only on inferring the subset of parameters which are "informed by data": adopting the Bayesian framework, we take this to mean the parameters which change significantly from the prior distribution to the posterior distribution. In this talk we will discuss a gradient-based construction of a low-dimensional approximation to the posterior distribution which codifies this core idea. In particular, our approximation is furnished with certifiable and computable error guarantees, which we obtain using tools from Markov semigroup theory such as the logarithmic Sobolev inequality and the Poincar\'e inequality. Joint work with Olivier Zahm and Youssef Marzouk.

This talk will report on several results, insights and perspectives Cyril Letrouit, Yury Polyanskiy, Philippe Rigollet and I have found regarding Transformers. We model Transformers as interacting particle systems (each particle representing a token), with a non-linear coupling called self-attention. When considering pure self-attention Transformers, we show that particles cluster in long time to different geometric configurations determined by spectral properties of the model weights. We also cover Transformers with layer normalization, which amounts to considering the interacting particle system on the unit sphere. On high-dimensional spheres, we prove that all randomly initialized particles converge to a single cluster. The result is made more precise by describing the precise phase transition between the clustering and non-clustering regimes. The appearance of metastability, and ideas for the low-dimensional regime, will be discussed.