Physics-informed Kernel Learning
Authors: Nathan Doumèche, Francis Bach, Gérard Biau, Claire Boyer
JMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time. |
| Researcher Affiliation | Collaboration | Nathan Doum eche EMAIL Sorbonne University, EDF R&D Francis Bach EMAIL INRIA, Ecole Normale Sup erieure, PSL University G erard Biau EMAIL Sorbonne University, Institut universitaire de France Claire Boyer EMAIL Universit e Paris-Saclay, Institut universitaire de France |
| Pseudocode | No | The paper describes the construction of the PIKL estimator and its algorithm using mathematical formulations and descriptions (e.g., Section 2: The PIKL Estimator, Section 3: The PIKL Algorithm in Practice), but it does not include a distinct, structured pseudocode or algorithm block. |
| Open Source Code | Yes | To enhance the reproducibility of our work, we provide a Python package that implements the PIKL estimator, designed to handle any linear PDE prior with constant coefficients in dimensions d = 1 and d = 2. This package is available at https://github.com/NathanDoumeche/numerical_PIML_kernel. Note that this package implements the matrix inversion of the PIKL formula (6) by solving a linear system using the LU decomposition. Of course, any other efficient method to avoid direct matrix inversion could be used instead, such as solving a linear system with the conjugate gradient method. |
| Open Datasets | No | The paper states: "To compare the PIKL and OLS estimators, we generate data such that Y = f (X) + ε, where X U(Ω), ε N(0, σ2)" and later mentions: "The training data set (Xi, Yi)1 i n is constructed such that...". This indicates that the authors generated their own synthetic datasets for the experiments and did not use or provide access to any pre-existing public datasets. |
| Dataset Splits | No | The paper describes how training data points are generated and assigned to different boundary conditions (e.g., "The training data set (Xi, Yi)1 i n is constructed such that if 1 i n/4 , then Xi = (0, Ui) and Yi = sin(πUi) + sin(4πUi)/2, if n/4 + 1 i 2 n/4 , then Xi = (Ui, 0) and Yi = 0... "). While it mentions evaluation on a "test set," it does not explicitly provide percentages or counts for training, validation, and test splits from a larger dataset, nor does it refer to standard predefined splits. |
| Hardware Specification | Yes | The training time for Vanilla PINNs is 7 minutes on an Nvidia L4 GPU (24 GB of RAM, 30.3 tera FLOPs for Float32). ... The training time for the PIKL estimator is 6 seconds on an Nvidia L4 GPU. |
| Software Dependencies | No | The paper mentions a "Python package" for implementation and discusses "Float32" and "Float64" precision. However, it does not provide specific version numbers for Python, any libraries, or other software dependencies. |
| Experiment Setup | Yes | To compare the PIKL and OLS estimators, we generate data such that Y = f (X) + ε, where X U(Ω), ε N(0, σ2) with σ = 0.5, and the target function is f = f1 (corresponding to (a1, a2) = (1, 0)). We implement the PIKL algorithm with 601 Fourier modes (m = 300) and s = 2. For the 1d-wave equation, it states: "We train our PIKL method using n = 105 data points and 1681 Fourier modes (i.e., m = 20)." For PINNs, it mentions: "The PINN is a fully-connected neural network with three hidden layers of size 10, using tanh as activation function, and optimized on 2 105 collocation points by 2000 gradient descent steps." |