相关论文: Optimal Prediction for Hamiltonian partial differe…
Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using…
Optimization constrained by high-fidelity computational models has potential for transformative impact. However, such optimization is frequently unattainable in practice due to the complexity and computational intensity of the model. An…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
A set of probabilistic predictions is well calibrated if the events that are predicted to occur with probability p do in fact occur about p fraction of the time. Well calibrated predictions are particularly important when machine learning…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
Conformal prediction offers finite-sample coverage guarantees under minimal assumptions. However, existing methods treat the entire modeling process as a black box, overlooking opportunities to exploit and understand modular structure. We…
We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
A popular approach for predicting the future of dynamical systems involves mapping them into a lower-dimensional "latent space" where prediction is easier. We show that the information-theoretically optimal approach uses different mappings…
We introduce the proximal optimal transport divergence, a novel discrepancy measure that interpolates between information divergences and optimal transport distances via an infimal convolution formulation. This divergence provides a…
Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…
Conformal prediction is a distribution-free and model-agnostic uncertainty-quantification method that provides finite-sample prediction intervals with guaranteed coverage. In this work, for the first time, we apply conformal-prediction to…
It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that standard approaches…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
We present a representation learning algorithm that learns a low-dimensional latent dynamical system from high-dimensional \textit{sequential} raw data, e.g., video. The framework builds upon recent advances in amortized inference methods…
This paper presents an iterative method suitable for inverting semilinear problems which are important kernels in many numerical applications. The primary idea is to employ a parametrization that is able to reduce semilinear problems into…
We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…