Turbulent combustion modeling presents a daunting challenge due to the complex interplay between two non-linear, multi-dimensional phenomena: combustion chemistry and turbulent dynamics. High-fidelity CFD simulations, albeit useful, are prohibitively computationally expensive. Consequently, data from such systems are often high-dimensional and limited in amount, making it difficult to draw insights. We propose an unsupervised generative learning method that simultaneously provides dimension reduction and clustering of complex, high-dimensional scientific data. Our model comprises a variational autoencoder with a Gaussian Mixture model in the latent space. During training, the model learns an evolving low-dimensional latent manifold of the data, facilitating disentanglement into clusters and yielding optimal dimension reduction.

We propose a general framework for high-dimensional regression problems, focusing on adaptive dimensionality reduction. The proposed framework approximates the target function by a mixture of experts model on a low-dimensional manifold, where each cluster is associated with a fixed-degree polynomial. We present a training strategy that leverages the expectation maximization (EM) algorithm. Under the probabilistic formulation, the EM step admits the form of embarrassingly parallelizable weighted least-squares solves.

Keywords: deep learning; dimensionality reduction; mixture of experts; nonparametric regression

Inverse problems in fluid dynamics are ubiquitous in science and engineering, with applications ranging from electronic cooling system design to ocean modeling. We propose a general and robust approach for solving inverse problems in the steady-state Navier-Stokes equations by combining deep neural networks and numerical partial differential equation (PDE) schemes. Our approach expresses numerical simulation as a computational graph with differentiable operators. We then solve inverse problems by constrained optimization, using gradients calculated from the computational graph with reverse-mode automatic differentiation.

Keywords: deep neural networks; inverse problems; numerical partial differential equations; finite element methods

Efficiently computing f(A)b, a function of a sparse Hermitian matrix times a vector, is an important component in numerous signal processing, machine learning, applied mathematics, and computer science tasks. We propose and investigate two new methods to approximate f(A)b for large, sparse, Hermitian matrices A. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both methods is to first estimate the spectral density of A, and then find polynomials of a fixed order that better approximate the function f on areas of the spectrum with a higher density of eigenvalues. 

Keywords: matrix function; spectral density estimation; polynomial approximation; orthogonal polynomials; graph spectral filtering

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