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Gaussian processes (GPs) based methods for solving partial differential equations (PDEs) demonstrate great promise by bridging the gap between the theoretical rigor of traditional numerical algorithms and the flexible design of machine…
The fast computation of large kernel sums is a challenging task, which arises as a subproblem in any kernel method. We approach the problem by slicing, which relies on random projections to one-dimensional subspaces and fast Fourier…
Deploying deep learning models in safety-critical applications remains a very challenging task, mandating the provision of assurances for the dependable operation of these models. Uncertainty quantification (UQ) methods estimate the model's…
We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…
This work introduces an end-to-end framework for multi-asset option pricing that combines market-consistent risk-neutral density recovery with quantum-accelerated numerical integration. We first calibrate arbitrage-free marginal…
Concept Bottleneck Model (CBM) is a methods for explaining neural networks. In CBM, concepts which correspond to reasons of outputs are inserted in the last intermediate layer as observed values. It is expected that we can interpret the…
We propose a general framework using spike-and-slab prior distributions to aid with the development of high-dimensional Bayesian inference. Our framework allows inference with a general quasi-likelihood function. We show that highly…
Differentiable programming has emerged as a key programming paradigm empowering rapid developments of deep learning while its applications to important computational methods such as Monte Carlo remain largely unexplored. Here we present the…
This paper provides a framework in which multilevel Monte Carlo and continuous level Monte Carlo can be compared. In continuous level Monte Carlo the level of refinement is determined by an exponentially distributed random variable, which…
As the amount and complexity of available data increases, the need for robust statistical learning becomes more pressing. To enhance resilience against model misspecification, the generalized posterior inference method adjusts the…
Quantum error mitigation (QEM) is a class of promising techniques for reducing the computational error of variational quantum algorithms. In general, the computational error reduction comes at the cost of a sampling overhead due to the…
The Diffusion Monte Carlo (DMC) method is applied to the water monomer, dimer, and hexamer, using q-TIP4P/F, one of the most simple, empirical water models with flexible monomers. The bias in the time step ($\Delta\tau$) and population size…
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it…
Estimating the predictive uncertainty of a Bayesian learning model is critical in various decision-making problems, e.g., reinforcement learning, detecting adversarial attack, self-driving car. As the model posterior is almost always…
We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous…
We establish epigraphical and uniform laws of large numbers for sample-based approximations of law invariant risk functionals. These sample-based approximation schemes include Monte Carlo (MC) and certain randomized quasi-Monte Carlo…
In the stochastic gradient descent (SGD) for sequential simulations such as the neural stochastic differential equations, the Multilevel Monte Carlo (MLMC) method is known to offer better theoretical computational complexity compared to the…
This paper introduces quasi-Monte Carlo latent variable models (QLVMs): a class of deep generative models that are specialized for finding extremely low-dimensional and interpretable embeddings of high-dimensional datasets. Unlike standard…
We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space $X$…
Discrepancies play an important role in the study of uniformity properties of point sets. Their probability distributions are a help in the analysis of the efficiency of the Quasi Monte Carlo method of numerical integration, which uses…