Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics

Physics-informed machine learning (PIML) as a means of solving partial differential equations (PDE) has garnered much attention in the Computational Science and Engineering (CS&E) world. This topic encompasses a broad array of methods and models aimed at solving a single or a collection of PDE problems, called multitask learning. PIML is characterized by the incorporation of physical laws into the training process of machine learning models in lieu of large data when solving PDE problems. Despite the overall success of this collection of methods, it remains incredibly difficult to analyze, benchmark, and generally compare one approach to another. Using Kolmogorov n-widths as a measure of effectiveness of approximating functions, we judiciously apply this metric in the comparison of various multitask PIML architectures. We compute lower accuracy bounds and analyze the model's learned basis functions on various PDE problems. This is the first objective metric for comparing multitask PIML architectures and helps remove uncertainty in model validation from selective sampling and overfitting. We also identify avenues of improvement for model architectures, such as the choice of activation function, which can drastically affect model generalization to "worst-case" scenarios, which is not observed when reporting task-specific errors. We also incorporate this metric into the optimization process through regularization, which improves the models' generalizability over the multitask PDE problem.

Diffeomorphic Measure Matching with Kernels for Generative Modeling

This article presents a general framework for the transport of probability measures towards minimum divergence generative modeling and sampling using ordinary differential equations (ODEs) and Reproducing Kernel Hilbert Spaces (RKHSs), inspired by ideas from diffeomorphic matching and image registration. A theoretical analysis of the proposed method is presented, giving a priori error bounds in terms of the complexity of the model, the number of samples in the training set, and model misspecification. An extensive suite of numerical experiments further highlights the properties, strengths, and weaknesses of the method and extends its applicability to other tasks, such as conditional simulation and inference.

Computational Hypergraph Discovery, a Gaussian Process framework for connecting the dots

Most scientific challenges can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. While most Computational Science and Engineering and Scientific Machine Learning challenges can be framed as Type 1 and Type 2 problems, many scientific problems can only be categorized as Type 3. Despite their prevalence, these Type 3 challenges have been largely overlooked due to their inherent complexity. Although Gaussian Process (GP) methods are sometimes perceived as well-founded but old technology limited to Type 1 curve fitting, their scope has recently been expanded to Type 2 problems. In this paper, we introduce an interpretable GP framework for Type 3 problems, targeting the data-driven discovery and completion of computational hypergraphs. Our approach is based on a kernel generalization of Row Echelon Form reduction from linear systems to nonlinear ones and variance-based analysis. Here, variables are linked via GPs and those contributing to the highest data variance unveil the hypergraph's structure. We illustrate the scope and efficiency of the proposed approach with applications to (algebraic) equation discovery, network discovery (gene pathways, chemical, and mechanical) and raw data analysis.

Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning

Machine Learning (ML) and Algorithmic Information Theory (AIT) look at Complexity from different points of view. We explore the interface between AIT and Kernel Methods (that are prevalent in ML) by adopting an AIT perspective on the problem of learning kernels from data, in kernel ridge regression, through the method of Sparse Kernel Flows. In particular, by looking at the differences and commonalities between Minimal Description Length (MDL) and Regularization in Machine Learning (RML), we prove that the method of Sparse Kernel Flows is the natural approach to adopt to learn kernels from data. This paper shows that it is not necessary to use the statistical route to derive Sparse Kernel Flows and that one can directly work with code-lengths and complexities that are concepts that show up in AIT.

Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

Kernel Methods are Competitive for Operator Learning

We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator $\mathcal{G}^\dagger\,:\, \mathcal{U}\to \mathcal{V}$ are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations $\phi(u_i), \varphi(v_i)$ of input/output functions $v_i=\mathcal{G}^\dagger(u_i)$ ($i=1,\ldots,N$), and the measurement operators $\phi\,:\, \mathcal{U}\to \mathbb{R}^n$ and $\varphi\,:\, \mathcal{V} \to \mathbb{R}^m$ are linear. Writing $\psi\,:\, \mathbb{R}^n \to \mathcal{U}$ and $\chi\,:\, \mathbb{R}^m \to \mathcal{V}$ for the optimal recovery maps associated with $\phi$ and $\varphi$, we approximate $\mathcal{G}^\dagger$ with $\bar{\mathcal{G}}=\chi \circ \bar{f} \circ \phi$ where $\bar{f}$ is an optimal recovery approximation of $f^\dagger:=\varphi \circ \mathcal{G}^\dagger \circ \psi\,:\,\mathbb{R}^n \to \mathbb{R}^m$. We show that, even when using vanilla kernels (e.g., linear or Mat\'{e}rn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.

Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes

We study the computational scalability of a Gaussian process (GP) framework for solving general nonlinear partial differential equations (PDEs). This framework transforms solving PDEs to solving quadratic optimization problem with nonlinear constraints. Its complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel of the GP and its partial derivatives at collocation points. We present a sparse Cholesky factorization algorithm for such kernel matrices based on the near-sparsity of the Cholesky factor under a new ordering of Diracs and derivative measurements. We rigorously identify the sparsity pattern and quantify the exponentially convergent accuracy of the corresponding Vecchia approximation of the GP, which is optimal in the Kullback-Leibler divergence. This enables us to compute $\epsilon$-approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log^d(N/\epsilon))$ in space and $O(N\log^{2d}(N/\epsilon))$ in time. With the sparse factors, gradient-based optimization methods become scalable. Furthermore, we can use the oftentimes more efficient Gauss-Newton method, for which we apply the conjugate gradient algorithm with the sparse factor of a reduced kernel matrix as a preconditioner to solve the linear system. We numerically illustrate our algorithm's near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Amp\`ere equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs.

Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems

Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a data-adapted kernel which can be learned by using Kernel Flows. The method of Kernel Flows is a trainable machine learning method that learns the optimal parameters of a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used. The objective function could be a short-term prediction or some other objective for other variants of Kernel Flows). However, this method is limited by the choice of the base kernel. In this paper, we introduce the method of \emph{Sparse Kernel Flows } in order to learn the ``best'' kernel by starting from a large dictionary of kernels. It is based on sparsifying a kernel that is a linear combination of elemental kernels. We apply this approach to a library of 132 chaotic systems.

Multiclass classification utilising an estimated algorithmic probability prior

Methods of pattern recognition and machine learning are applied extensively in science, technology, and society. Hence, any advances in related theory may translate into large-scale impact. Here we explore how algorithmic information theory, especially algorithmic probability, may aid in a machine learning task. We study a multiclass supervised classification problem, namely learning the RNA molecule sequence-to-shape map, where the different possible shapes are taken to be the classes. The primary motivation for this work is a proof of concept example, where a concrete, well-motivated machine learning task can be aided by approximations to algorithmic probability. Our approach is based on directly estimating the class (i.e., shape) probabilities from shape complexities, and using the estimated probabilities as a prior in a Gaussian process learning problem. Naturally, with a large amount of training data, the prior has no significant influence on classification accuracy, but in the very small training data regime, we show that using the prior can substantially improve classification accuracy. To our knowledge, this work is one of the first to demonstrate how algorithmic probability can aid in a concrete, real-world, machine learning problem.

One-Shot Learning of Stochastic Differential Equations with Computational Graph Completion

We consider the problem of learning Stochastic Differential Equations of the form $dX_t = f(X_t)dt+\sigma(X_t)dW_t $ from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions $f$, $\sigma$, and stochastic process $dW_t$ representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a simple kernel-based solution to this problem that can be decomposed as follows: (1) Represent the time-increment map $X_t \rightarrow X_{t+dt}$ as a Computational Graph in which $f$, $\sigma$ and $dW_t$ appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.