Gaussian Process Approximation for Differential Equations
(2026) In Bachelor’s Theses in Mathematical Sciences NUMK11 20252Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- This thesis develops a Bayesian framework based on Gaussian processes for inferring parameters of differential operators from data and reconstructing the corresponding solution together with its uncertainty. Such inverse problems are often ill-posed and sensitive to noise, which makes classical deterministic inverse approaches unstable and limits their ability to quantify uncertainty. The setup considers an unknown function u(x) and a forcing term f(x) related through a differential operator acting on u(x). The proposed approach avoids the explicit numerical solution of differential equations by constructing a Gaussian process approximation of the unknown functions. Gaussian process priors are used to encode smoothness assumptions over the... (More)
- This thesis develops a Bayesian framework based on Gaussian processes for inferring parameters of differential operators from data and reconstructing the corresponding solution together with its uncertainty. Such inverse problems are often ill-posed and sensitive to noise, which makes classical deterministic inverse approaches unstable and limits their ability to quantify uncertainty. The setup considers an unknown function u(x) and a forcing term f(x) related through a differential operator acting on u(x). The proposed approach avoids the explicit numerical solution of differential equations by constructing a Gaussian process approximation of the unknown functions. Gaussian process priors are used to encode smoothness assumptions over the input domain. This enables estimation of the operator parameters by minimizing the negative marginal likelihood of the Gaussian process model. Using the prior together with the data likelihood, a posterior distribution over the functions is obtained. The posterior distribution provides solutions to the differential equation through point estimates and associated measures of uncertainty. The methodology is applied to several differential equations, including the heat equation, an integro-differential equation and the Allen–Cahn reaction-diffusion equation. The parameters are inferred from data. The numerical experiments show that, for the linear test cases, the operator parameters can be recovered with high accuracy and the corresponding solutions can be reconstructed reliably, even when the data are scarce and noisy. The results improve as the amount of training data increases and deteriorate as the noise level increases. For the nonlinear Allen–Cahn reaction-diffusion equation, the method remains applicable, but the parameter estimates and reconstructions are less accurate and require substantially more training data. (Less)
- Popular Abstract
- Governing laws of nature evolve continuously, and in order to mathematically capture this, they are described using differential equations. These equations describe how quantities change with respect to variables such as time or space. By finding a solution to a differential equation, one is able to predict the behaviour of the physical system. To make this concrete, consider the following example of when a differential equation, the heat equation, is used and what knowledge is gained by solving it:
A medical treatment for cancer is to insert a small heating device within a tumour to destroy the malignant tissue cells. The procedure needs to be precise so that the heat spreading from the device removes the tumour while not damaging the... (More) - Governing laws of nature evolve continuously, and in order to mathematically capture this, they are described using differential equations. These equations describe how quantities change with respect to variables such as time or space. By finding a solution to a differential equation, one is able to predict the behaviour of the physical system. To make this concrete, consider the following example of when a differential equation, the heat equation, is used and what knowledge is gained by solving it:
A medical treatment for cancer is to insert a small heating device within a tumour to destroy the malignant tissue cells. The procedure needs to be precise so that the heat spreading from the device removes the tumour while not damaging the surrounding healthy tissue. We already know the general equation for how heat spreads, but not the specific parameters that depend on the material through which the heat is diffusing. Furthermore, we are only able to measure a few temperature values at certain locations and times in the tissue, as well as the temperature of the heating device.
Given this situation, the method used in this thesis, called the Gaussian process, allows us to use this limited information to infer the specific parameters of the heat equation in the tissue and to reconstruct the temperature at any other location and time. This reconstruction of the temperature field is what it means to solve the differential equation. More specifically, the method applying the Gaussian process models both the unknown temperature field and the unknown heat source of the device as a probability distribution over functions. Prior assumptions favouring reasonable solution behaviour are combined with the measurement points to form a new distribution, called the posterior, over the possible temperature functions. Because Gaussian processes are governed by Gaussian distributions, each posterior is completely described by its mean and variance. In this setting, we obtain a posterior mean and variance for the temperature function and another for the transmitter (heat-source) function. These means give the inferred temperature and the inferred heat source as functions of space and time, while the corresponding variances quantify how uncertain these inferred functions are. This provides an accurate understanding of how heat from the device spreads through the tissue over time. Another strength of this method is that even though only a few measurements are available and these measurements are noisy, the resulting predictions remain highly accurate. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/student-papers/record/9225443
- author
- Ben Moussa, Noah LU
- supervisor
- organization
- alternative title
- Approximation av Differentialekvationer med Gaussprocesser
- course
- NUMK11 20252
- year
- 2026
- type
- M2 - Bachelor Degree
- subject
- keywords
- Gaussian process, Uncertainty quantification, Bayesian inference, Differential equation, Parameter estimation
- publication/series
- Bachelor’s Theses in Mathematical Sciences
- report number
- LUNFNA-4068-2026
- ISSN
- 1654-6229
- other publication id
- 2026:K3
- language
- English
- id
- 9225443
- date added to LUP
- 2026-05-07 13:11:15
- date last changed
- 2026-05-07 13:11:15
@misc{9225443,
abstract = {{This thesis develops a Bayesian framework based on Gaussian processes for inferring parameters of differential operators from data and reconstructing the corresponding solution together with its uncertainty. Such inverse problems are often ill-posed and sensitive to noise, which makes classical deterministic inverse approaches unstable and limits their ability to quantify uncertainty. The setup considers an unknown function u(x) and a forcing term f(x) related through a differential operator acting on u(x). The proposed approach avoids the explicit numerical solution of differential equations by constructing a Gaussian process approximation of the unknown functions. Gaussian process priors are used to encode smoothness assumptions over the input domain. This enables estimation of the operator parameters by minimizing the negative marginal likelihood of the Gaussian process model. Using the prior together with the data likelihood, a posterior distribution over the functions is obtained. The posterior distribution provides solutions to the differential equation through point estimates and associated measures of uncertainty. The methodology is applied to several differential equations, including the heat equation, an integro-differential equation and the Allen–Cahn reaction-diffusion equation. The parameters are inferred from data. The numerical experiments show that, for the linear test cases, the operator parameters can be recovered with high accuracy and the corresponding solutions can be reconstructed reliably, even when the data are scarce and noisy. The results improve as the amount of training data increases and deteriorate as the noise level increases. For the nonlinear Allen–Cahn reaction-diffusion equation, the method remains applicable, but the parameter estimates and reconstructions are less accurate and require substantially more training data.}},
author = {{Ben Moussa, Noah}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor’s Theses in Mathematical Sciences}},
title = {{Gaussian Process Approximation for Differential Equations}},
year = {{2026}},
}