# Fenrir: Physics-Enhanced Regression for Initial Value Problems

arXiv: https://arxiv.org/abs/2202.01287 · HTML: https://mlsanity.com/paper/2202.01287/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 71 references that are in the map.
- First posted on arXiv 2 Feb 2022.

## Builds on (most cited first, 20 of 20)

- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Calibrated Adaptive Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.08202/) (2020)
- [A Probabilistic State Space Model for Joint Inference from Differential Equations and Data](https://mlsanity.com/paper/2103.10153/) (2021)
- [Bayesian ODE Solvers: The Maximum A Posteriori Estimate](https://mlsanity.com/paper/2004.00623/) (2020)
- [Stable Implementation of Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.10106/) (2020)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Fast Gaussian Process Based Gradient Matching for Parameter Identification in Systems of Nonlinear ODEs](https://mlsanity.com/paper/1804.04378/) (2018)
- [ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systems](https://mlsanity.com/paper/1902.06278/) (2019)
- [Convergence Rates of Gaussian ODE Filters](https://mlsanity.com/paper/1807.09737/) (2018)
- [Differentiable Likelihoods for Fast Inversion of 'Likelihood-Free' Dynamical Systems](https://mlsanity.com/paper/2002.09301/) (2020)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [Scalable Variational Inference for Dynamical Systems](https://mlsanity.com/paper/1705.07079/) (2017)
- [Pick-and-Mix Information Operators for Probabilistic ODE Solvers](https://mlsanity.com/paper/2110.10770/) (2021)
- [Laplace based approximate posterior inference for differential equation models](https://mlsanity.com/paper/1607.07203/) (2016)
- [Implicit Probabilistic Integrators for ODEs](https://mlsanity.com/paper/1805.07970/) (2018)
- [Bayesian Filtering for ODEs with Bounded Derivatives](https://mlsanity.com/paper/1709.08471/) (2017)
- [Estimation of ordinary differential equation models with discretization error quantification](https://mlsanity.com/paper/1907.10565/) (2019)

## Built on by (most cited first, 11 of 11)

- [Random Grid Neural Processes for Parametric Partial Differential Equations](https://mlsanity.com/paper/2301.11040/) (2023)
- [The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions](https://mlsanity.com/paper/2306.07774/) (2023)
- [Probabilistic Exponential Integrators](https://mlsanity.com/paper/2305.14978/) (2023)
- [Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations](https://mlsanity.com/paper/2306.05566/) (2023)
- [Parallel-in-Time Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2310.01145/) (2023)
- [Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations](https://mlsanity.com/paper/2402.12231/) (2024)
- [Uncertainty Informed Optimal Resource Allocation with Gaussian Process based Bayesian Inference](https://mlsanity.com/paper/2307.00032/) (2023)
- [Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements](https://mlsanity.com/paper/2410.10530/) (2024)
- [Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2503.04684/) (2025)
- [Numerically robust Gaussian state estimation with singular observation noise](https://mlsanity.com/paper/2503.10279/) (2025)
- [Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs](https://mlsanity.com/paper/2606.08203/) (2026)

## Abstract (arXiv)

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. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.