# Probabilistic Gradients for Fast Calibration of Differential Equation Models

arXiv: https://arxiv.org/abs/2009.04239 · HTML: https://mlsanity.com/paper/2009.04239/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Sep 2020.

## Built on by (most cited first, 3 of 3)

- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [Learning to Solve Related Linear Systems](https://mlsanity.com/paper/2503.17265/) (2025)

## Abstract (arXiv)

Calibration of large-scale differential equation models to observational or experimental data is a widespread challenge throughout applied sciences and engineering. A crucial bottleneck in state-of-the art calibration methods is the calculation of local sensitivities, i.e. derivatives of the loss function with respect to the estimated parameters, which often necessitates several numerical solves of the underlying system of partial or ordinary differential equations. In this paper we present a new probabilistic approach to computing local sensitivities. The proposed method has several advantages over classical methods. Firstly, it operates within a constrained computational budget and provides a probabilistic quantification of uncertainty incurred in the sensitivities from this constraint. Secondly, information from previous sensitivity estimates can be recycled in subsequent computations, reducing the overall computational effort for iterative gradient-based calibration methods. The methodology presented is applied to two challenging test problems and compared against classical methods.