On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are particularly well suited to weakening the effect of the curse of dimensionality, a term coined by Richard E. Bellman in the late `50s to describe challenges such as the exponential dependence of the sample complexity, i.e., the number of samples required to solve an approximation problem, on the dimension of the ambient space. However, although DNNs have been used to solve PDEs since the `90s, the literature underpinning their mathematical efficiency in terms of numerical analysis (i.e., stability, accuracy, and sample complexity), is only recently beginning to emerge. In this paper, we leverage recent advancements in function approximation using sparsity-based techniques and random sampling to develop and analyze an efficient high-dimensional PDE solver based on DL. We show, both theoretically and numerically, that it can compete with a novel stable and accurate compressive spectral collocation method for the solution of high-dimensional, steady-state diffusion-reaction equations with periodic boundary conditions. In particular, we demonstrate a new practical existence theorem, which establishes the existence of a class of trainable DNNs with suitable bounds on the network architecture and a sufficient condition on the sample complexity, with logarithmic or, at worst, linear scaling in dimension, such that the resulting networks stably and accurately approximate a diffusion-reaction PDE with high probability.

This work considers the fundamental problem of learning an unknown object from training data using a given model class. We introduce a framework that allows for objects in arbitrary Hilbert spaces, general types of (random) linear measurements as training data and general types of nonlinear model classes. We establish a series of learning guarantees for this framework, which provide explicit relations between the amount of training data and the model class to ensure near-best generalization bounds. In doing so, we introduce the key notion of the variation of a model class with respect to a distribution of sampling operators. We show that this framework can accommodate many different types of well-known problems of interest, such as matrix sketching by random sampling, compressed sensing with isotropic vectors, active learning in regression and compressed sensing with generative models. In all cases, known results become straightforward corollaries of our general theory. Hence, this work provides a powerful framework for studying and analyzing many different types of learning problems.

Operator learning problems arise in many key areas of scientific computing where Partial Differential Equations (PDEs) are used to model physical systems. In such scenarios, the operators map between Banach or Hilbert spaces. In this work, we tackle the problem of learning operators between Banach spaces, in contrast to the vast majority of past works considering only Hilbert spaces. We focus on learning holomorphic operators -- an important class of problems with many applications. We combine arbitrary approximate encoders and decoders with standard feedforward Deep Neural Network (DNN) architectures -- specifically, those with constant width exceeding the depth -- under standard $\ell^2$-loss minimization. We first identify a family of DNNs such that the resulting Deep Learning (DL) procedure achieves optimal generalization bounds for such operators. For standard fully-connected architectures, we then show that there are uncountably many minimizers of the training problem that yield equivalent optimal performance. The DNN architectures we consider are `problem agnostic', with width and depth only depending on the amount of training data $m$ and not on regularity assumptions of the target operator. Next, we show that DL is optimal for this problem: no recovery procedure can surpass these generalization bounds up to log terms. Finally, we present numerical results demonstrating the practical performance on challenging problems including the parametric diffusion, Navier-Stokes-Brinkman and Boussinesq PDEs.

We introduce a general framework for active learning in regression problems. Our framework extends the standard setup by allowing for general types of data, rather than merely pointwise samples of the target function. This generalization covers many cases of practical interest, such as data acquired in transform domains (e.g., Fourier data), vector-valued data (e.g., gradient-augmented data), data acquired along continuous curves, and, multimodal data (i.e., combinations of different types of measurements). Our framework considers random sampling according to a finite number of sampling measures and arbitrary nonlinear approximation spaces (model classes). We introduce the concept of \textit{generalized Christoffel functions} and show how these can be used to optimize the sampling measures. We prove that this leads to near-optimal sample complexity in various important cases. This paper focuses on applications in scientific computing, where active learning is often desirable, since it is usually expensive to generate data. We demonstrate the efficacy of our framework for gradient-augmented learning with polynomials, Magnetic Resonance Imaging (MRI) using generative models and adaptive sampling for solving PDEs using Physics-Informed Neural Networks (PINNs).

In this paper, we consider the use of total variation (TV) minimization for compressive imaging, that is, image reconstruction from subsampled measurements. Focusing on two important imaging modalities---namely, Fourier imaging and structured binary imaging via the Walsh--Hadamard transform---we derive uniform recovery guarantees asserting stable and robust recovery for arbitrary random sampling strategies. Using this, we then derive a class of sampling strategies which are theoretically near-optimal for recovery of approximately gradient-sparse images. For Fourier sampling, we show recovery of such an image from $m \gtrsim_d s \cdot \log^2(s) \cdot \log^4(N)$ measurements, in $d \geq 1$ dimensions. When $d = 2$, this improves the current state-of-the-art result by a factor of $\log(s) \cdot \log(N)$. It also extends it to arbitrary dimensions $d \geq 2$. For Walsh sampling, we prove that $m \gtrsim_d s \cdot \log^2(s) \cdot \log^2(N/s) \cdot \log^3(N) $ measurements suffice in $d \geq 2$ dimensions. To the best of our knowledge, this is the first recovery guarantee for structured binary sampling with TV minimization.

Motivation: The prediction of drug resistance and the identification of its mechanisms in bacteria such as Mycobacterium tuberculosis, the etiological agent of tuberculosis, is a challenging problem. Modern methods based on testing against a catalogue of previously identified mutations often yield poor predictive performance. On the other hand, machine learning techniques have demonstrated high predictive accuracy, but many of them lack interpretability to aid in identifying specific mutations which lead to resistance. We propose a novel technique, inspired by the group testing problem and Boolean compressed sensing, which yields highly accurate predictions and interpretable results at the same time. Results: We develop a modified version of the Boolean compressed sensing problem for identifying drug resistance, and implement its formulation as an integer linear program. This allows us to characterize the predictive accuracy of the technique and select an appropriate metric to optimize. A simple adaptation of the problem also allows us to quantify the sensitivity-specificity trade-off of our model under different regimes. We test the predictive accuracy of our approach on a variety of commonly used antibiotics in treating tuberculosis and find that it has accuracy comparable to that of standard machine learning models and points to several genes with previously identified association to drug resistance.