Xianjin Yang

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.

Xianjin Yang, Matthieu Darcy, Matthew Hudes et al.

We present an operator learning framework for solving non-perturbative functional renormalization group equations, which are integro-differential equations defined on functionals. Our proposed approach uses Gaussian process operator learning to construct a flexible functional representation formulated directly on function space, making it independent of a particular equation or discretization. Our method is flexible, and can apply to a broad range of functional differential equations while still allowing for the incorporation of physical priors in either the prior mean or the kernel design. We demonstrate the performance of our method on several relevant equations, such as the Wetterich and Wilson--Polchinski equations, showing that it achieves equal or better performance than existing approximations such as the local-potential approximation, while being significantly more flexible. In particular, our method can handle non-constant fields, making it promising for the study of more complex field configurations, such as instantons.

Jinyan Guo, Chenchen Mou, Xianjin Yang et al.

This paper presents a Gaussian Process (GP) framework, a non-parametric technique widely acknowledged for regression and classification tasks, to address inverse problems in mean field games (MFGs). By leveraging GPs, we aim to recover agents' strategic actions and the environment's configurations from partial and noisy observations of the population of agents and the setup of the environment. Our method is a probabilistic tool to infer the behaviors of agents in MFGs from data in scenarios where the comprehensive dataset is either inaccessible or contaminated by noises.

Jingguo Zhang, Xianjin Yang, Chenchen Mou et al.

Mean field games (MFGs) describe the collective behavior of large populations of interacting agents. In this work, we tackle ill-posed inverse problems in potential MFGs, aiming to recover the agents' population, momentum, and environmental setup from limited, noisy measurements and partial observations. These problems are ill-posed because multiple MFG configurations can explain the same data, or different parameters can yield nearly identical observations. Nonetheless, they remain crucial in practice for real-world scenarios where data are inherently sparse or noisy, or where the MFG structure is not fully determined. Our focus is on finding surrogate MFGs that accurately reproduce the observed data despite these challenges. We propose two Gaussian process (GP)-based frameworks: an inf-sup formulation and a bilevel approach. The choice between them depends on whether the unknown parameters introduce concavity in the objective. In the inf-sup framework, we use the linearity of GPs and their parameterization structure to maintain convex-concave properties, allowing us to apply standard convex optimization algorithms. In the bilevel framework, we employ a gradient-descent-based algorithm and introduce two methods for computing the outer gradient. The first method leverages an existing solver for the inner potential MFG and applies automatic differentiation, while the second adopts an adjoint-based strategy that computes the outer gradient independently of the inner solver. Our numerical experiments show that when sufficient prior information is available, the unknown parameters can be accurately recovered. Otherwise, if prior information is limited, the inverse problem is ill-posed, but our frameworks can still produce surrogate MFG models that closely match observed data.

Xianjin Yang, Jingguo Zhang

We propose a Gaussian Process (GP)-based policy iteration framework for addressing both forward and inverse problems in Hamilton--Jacobi--Bellman (HJB) equations and mean field games (MFGs). Policy iteration is formulated as an alternating procedure between solving the value function under a fixed control policy and updating the policy based on the resulting value function. By exploiting the linear structure of GPs for function approximation, each policy evaluation step admits an explicit closed-form solution, eliminating the need for numerical optimization. To improve convergence, we incorporate the additive Schwarz acceleration as a preconditioning step following each policy update. Numerical experiments demonstrate the effectiveness of Schwarz acceleration in improving computational efficiency.

Ricardo Baptista, Edoardo Calvello, Matthieu Darcy et al.

We consider the use of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically approximate the solutions to nonlinear partial differential equations (PDEs) with rough forcing or source terms, which commonly arise as pathwise solutions to stochastic PDEs. Kernel methods have recently been generalized to solve nonlinear PDEs by approximating their solutions as the maximum a posteriori estimator of GPs that are conditioned to satisfy the PDE at a finite set of collocation points. The convergence and error guarantees of these methods, however, rely on the PDE being defined in a classical sense and its solution possessing sufficient regularity to belong to the associated reproducing kernel Hilbert space. We propose a generalization of these methods to handle roughly forced nonlinear PDEs while preserving convergence guarantees with an oversmoothing GP kernel that is misspecified relative to the true solution's regularity. This is achieved by conditioning a regular GP to satisfy the PDE with a modified source term in a weak sense (when integrated against a finite number of test functions). This is equivalent to replacing the empirical $L^2$-loss on the PDE constraint by an empirical negative-Sobolev norm. We further show that this loss function can be used to extend physics-informed neural networks (PINNs) to stochastic equations, thereby resulting in a new NN-based variant termed Negative Sobolev Norm-PINN (NeS-PINN).

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.

Nicholas H. Nelsen, Houman Owhadi, Andrew M. Stuart et al.

Methods for solving scientific computing and inference problems, such as kernel- and neural network-based approaches for partial differential equations (PDEs), inverse problems, and supervised learning tasks, depend crucially on the choice of hyperparameters. Specifically, the efficacy of such methods, and in particular their accuracy, stability, and generalization properties, strongly depends on the choice of hyperparameters. While bilevel optimization offers a principled framework for hyperparameter tuning, its nested optimization structure can be computationally demanding, especially in PDE-constrained contexts. In this paper, we propose an efficient strategy for hyperparameter optimization within the bilevel framework by employing a Gauss-Newton linearization of the inner optimization step. Our approach provides closed-form updates, eliminating the need for repeated costly PDE solves. As a result, each iteration of the outer loop reduces to a single linearized PDE solve, followed by explicit gradient-based hyperparameter updates. We demonstrate the effectiveness of the proposed method through Gaussian process models applied to nonlinear PDEs and to PDE inverse problems. Extensive numerical experiments highlight substantial improvements in accuracy and robustness compared to conventional random hyperparameter initialization. In particular, experiments with additive kernels and neural network-parameterized deep kernels demonstrate the method's scalability and effectiveness for high-dimensional hyperparameter optimization.