Mercer's expansion and Mercer's theorem are cornerstone results in kernel theory. While the classical Mercer's theorem only considers continuous symmetric positive definite kernels, analogous expansions are effective in practice for indefinite and asymmetric kernels. In this paper we extend Mercer's expansion to continuous kernels, providing a rigorous theoretical underpinning for indefinite and asymmetric kernels. We begin by demonstrating that Mercer's expansion may not be pointwise convergent for continuous indefinite kernels, before proving that the expansion of continuous kernels with bounded variation uniformly in each variable separably converges pointwise almost everywhere, almost uniformly, and unconditionally almost everywhere. We also describe an algorithm for computing Mercer's expansion for general kernels and give new decay bounds on its terms.

We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of recovering the solution operator of hyperbolic PDEs is the presence of characteristics, along which the associated Green's function is discontinuous. Therefore, a central component of our algorithm is a rank detection scheme that identifies the approximate location of the characteristics. By combining the randomized singular value decomposition with an adaptive hierarchical partition of the domain, we construct an approximant to the solution operator using $O(\Psi_\epsilon^{-1}\epsilon^{-7}\log(\Xi_\epsilon^{-1}\epsilon^{-1}))$ input-output pairs with relative error $O(\Xi_\epsilon^{-1}\epsilon)$ in the operator norm as $\epsilon\to0$, with high probability. Here, $\Psi_\epsilon$ represents the existence of degenerate singular values of the solution operator, and $\Xi_\epsilon$ measures the quality of the training data. Our assumptions on the regularity of the coefficients of the hyperbolic PDE are relatively weak given that hyperbolic PDEs do not have the "instantaneous smoothing effect" of elliptic and parabolic PDEs, and our recovery rate improves as the regularity of the coefficients increases. We also include numerical experiments which corroborate our theoretical findings.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint. In this paper, we partially explain this mystery by proving that without querying the adjoint, one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis. We then apply the result to recovering Green's functions of elliptic partial differential operators and derive an adjoint-free sample complexity bound. While existing theory justifies low sample complexity in operator learning, ours is the first adjoint-free analysis that attempts to close the gap between theory and practice.

Many real-world datasets contain missing entries and mixed data types including categorical and ordered (e.g. continuous and ordinal) variables. Imputing the missing entries is necessary, since many data analysis pipelines require complete data, but challenging especially for mixed data. This paper proposes a probabilistic imputation method using an extended Gaussian copula model that supports both single and multiple imputation. The method models mixed categorical and ordered data using a latent Gaussian distribution. The unordered characteristics of categorical variables is explicitly modeled using the argmax operator. The method makes no assumptions on the data marginals nor does it require tuning any hyperparameters. Experimental results on synthetic and real datasets show that imputation with the extended Gaussian copula outperforms the current state-of-the-art for both categorical and ordered variables in mixed data.

Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with partial differential equations (PDEs) from pairs of forcing functions and solutions when numerical solvers are not available or the underlying physics is poorly understood. In this work, we attempt to provide theoretical foundations to understand the amount of training data needed to learn time-dependent PDEs. Given input-output pairs from a parabolic PDE in any spatial dimension $n\geq 1$, we derive the first theoretically rigorous scheme for learning the associated solution operator, which takes the form of a convolution with a Green's function $G$. Until now, rigorously learning Green's functions associated with time-dependent PDEs has been a major challenge in the field of scientific machine learning because $G$ may not be square-integrable when $n>1$, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of $G$ together with randomized numerical linear algebra, we construct an approximant to $G$ that achieves a relative error of $\smash{\mathcal{O}(\Gamma_\epsilon^{-1/2}\epsilon)}$ in the $L^1$-norm with high probability by using at most $\smash{\mathcal{O}(\epsilon^{-\frac{n+2}{2}}\log(1/\epsilon))}$ input-output training pairs, where $\Gamma_\epsilon$ is a measure of the quality of the training dataset for learning $G$, and $\epsilon>0$ is sufficiently small.

We consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. We establish optimal bounds in terms of network complexity and prove that rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth. The flexibility and smoothness of rational activation functions make them an attractive alternative to ReLU, as we demonstrate with numerical experiments.