Contributions: In this work, we develop a projected Stein variational Newton method (pSVN) to tackle the challenge of high-dimensional Bayesian inference by exploiting the intrinsic lowdimensional geometric structure of the posterior distribution (where it departs from the prior), as characterized by the dominant spectrum of the prior-preconditioned Hessian of the negative log likelihood.

We propose a projected Stein variational Newton (pSVN) method for high-dimensional Bayesian inference.

We propose derivative-informed neural operators (DINOs), a general family of neural networks to approximate operators as infinite-dimensional mappings from input function spaces to output function spaces or quantities of interest. After discretizations both inputs and outputs are high-dimensional. We aim to approximate not only the operators with improved accuracy but also their derivatives (Jacobians) with respect to the input function-valued parameter to empower derivative-based algorithms in many applications, e.g., Bayesian inverse problems, optimization under parameter uncertainty, and optimal experimental design. The major difficulties include the computational cost of generating derivative training data and the high dimensionality of the problem leading to large training cost. To address these challenges, we exploit the intrinsic low-dimensionality of the derivatives and develop algorithms for compressing derivative information and efficiently imposing it in neural operator training yielding derivative-informed neural operators. We demonstrate that these advances can significantly reduce the costs of both data generation and training for large classes of problems (e.g., nonlinear steady state parametric PDE maps), making the costs marginal or comparable to the costs without using derivatives, and in particular independent of the discretization dimension of the input and output functions. Moreover, we show that the proposed DINO achieves significantly higher accuracy than neural operators trained without derivative information, for both function approximation and derivative approximation (e.g., Gauss-Newton Hessian), especially when the training data are limited.

We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces the long-standing curse of dimensionality. We overcome this challenge by exploiting the intrinsic low-rank structure in the difference between the posterior and prior distributions. The parameters are projected into a low-dimensional subspace to alleviate the approximation error of KDE in high dimensions. We formulate a projected Wasserstein gradient flow and analyze its convergence property under mild assumptions. Several numerical experiments illustrate the accuracy, convergence, and complexity scalability of pWGD with respect to parameter dimension, sample size, and processor cores.

The curse of dimensionality is a critical challenge in Bayesian inference for high dimensional parameters. In this work, we address this challenge by developing a projected Stein variational gradient descent (pSVGD) method, which projects the parameters into a subspace that is adaptively constructed using the gradient of the log-likelihood, and applies SVGD for the much lower-dimensional coefficients of the projection. We provide an upper bound for the projection error with respect to the posterior and demonstrate the accuracy (compared to SVGD) and scalability of pSVGD with respect to the number of parameters, samples, data points, and processor cores.