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Found 14 papers, 7 papers with code
Parallel-in-Time Probabilistic Numerical ODE Solvers
Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation.
The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions
In this paper, we propose a novel approximate Gaussian filtering and smoothing method which propagates low-rank approximations of the covariance matrices.
Probabilistic Exponential Integrators
However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability.
Orthonormal Expansions for Translation-Invariant Kernels
We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$.
Fenrir: Physics-Enhanced Regression for Initial Value Problems
We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem.
Pick-and-Mix Information Operators for Probabilistic ODE Solvers
Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference.
A probabilistic Taylor expansion with Gaussian processes
We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order.
Calibrated Adaptive Probabilistic ODE Solvers
The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error.
Continuous-Discrete Filtering and Smoothing on Submanifolds of Euclidean Space
For approximate filtering and smoothing the projection approach is taken, where it turns out that the prediction and smoothing equations are the same as in the case when the state variable evolves in Euclidean space.
Bayesian ODE Solvers: The Maximum A Posteriori Estimate
The remaining three classes are termed explicit, semi-implicit, and implicit, which are in similarity with the classical notions corresponding to conditions on the vector field, under which the filter update produces a local maximum a posteriori estimate.
Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions
We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model in the sense that the model can become "slowly" overconfident at worst, regardless of the difference between the smoothness of the data-generating function and that expected by the model.
Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions.
Gaussian process classification using posterior linearisation
This paper proposes a new algorithm for Gaussian process classification based on posterior linearisation (PL).
Student-t Process Quadratures for Filtering of Non-Linear Systems with Heavy-Tailed Noise
Advantage of the Student- t process quadrature over the traditional Gaussian process quadrature, is that the integral variance depends also on the function values, allowing for a more robust modelling of the integration error.
