Théo Bourdais

Théo Bourdais, Pau Batlle, Xianjin Yang et al.

Most problems within and beyond the scientific domain 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. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.

Théo Bourdais, Houman Owhadi

Whether deterministic or stochastic, models can be viewed as functions designed to approximate a specific quantity of interest. We introduce Minimal Empirical Variance Aggregation (MEVA), a data-driven framework that integrates predictions from various models, enhancing overall accuracy by leveraging the individual strengths of each. This non-intrusive, model-agnostic approach treats the contributing models as black boxes and accommodates outputs from diverse methodologies, including machine learning algorithms and traditional numerical solvers. We advocate for a point-wise linear aggregation process and consider two methods for optimizing this aggregate: Minimal Error Aggregation (MEA), which minimizes the prediction error, and Minimal Variance Aggregation (MVA), which focuses on reducing variance. We prove a theorem showing that MVA can be more robustly estimated from data than MEA, making MEVA superior to Minimal Empirical Error Aggregation (MEEA). Unlike MEEA, which interpolates target values directly, MEVA formulates aggregation as an error estimation problem, which can be performed using any backbone learning paradigm. We demonstrate the versatility and effectiveness of our framework across various applications, including data science and partial differential equations, illustrating its ability to significantly enhance both robustness and accuracy.

Aras Bacho, Aleksei G. Sorokin, Xianjin Yang et al.

Neural operator learning methods have garnered significant attention in scientific computing for their ability to approximate infinite-dimensional operators. However, increasing their complexity often fails to substantially improve their accuracy, leaving them on par with much simpler approaches such as kernel methods and more traditional reduced-order models. In this article, we set out to address this shortcoming and introduce CHONKNORIS (Cholesky Newton--Kantorovich Neural Operator Residual Iterative System), an operator learning paradigm that can achieve machine precision. CHONKNORIS draws on numerical analysis: many nonlinear forward and inverse PDE problems are solvable by Newton-type methods. Rather than regressing the solution operator itself, our method regresses the Cholesky factors of the elliptic operator associated with Tikhonov-regularized Newton--Kantorovich updates. The resulting unrolled iteration yields a neural architecture whose machine-precision behavior follows from achieving a contractive map, requiring far lower accuracy than end-to-end approximation of the solution operator. We benchmark CHONKNORIS on a range of nonlinear forward and inverse problems, including a nonlinear elliptic equation, Burgers' equation, a nonlinear Darcy flow problem, the Calderón problem, an inverse wave scattering problem, and a problem from seismic imaging. We also present theoretical guarantees for the convergence of CHONKNORIS in terms of the accuracy of the emulated Cholesky factors. Additionally, we introduce a foundation model variant, FONKNORIS (Foundation Newton--Kantorovich Neural Operator Residual Iterative System), which aggregates multiple pre-trained CHONKNORIS experts for diverse PDEs to emulate the solution map of a novel nonlinear PDE. Our FONKNORIS model is able to accurately solve unseen nonlinear PDEs such as the Klein--Gordon and Sine--Gordon equations.