Notice: The reproducibility variables underlying each score are classified using an automated LLM-based pipeline, validated against a manually labeled dataset. LLM-based classification introduces uncertainty and potential bias; scores should be interpreted as estimates. Full accuracy metrics and methodology are described in Coakley et alK. L. Coakley, T. Snelleman, H. Hoos, and O. E. Gundersen, "The embrace of open science: An analysis of a decade of AI research and 56 800 conference papers," Under Review, 2026..
UEPI: Universal Energy-Behavior-Preserving Integrators for Energy Conservative/Dissipative Differential Equations
Authors: Elena Celledoni, Brynjulf Owren, Chong Shen, Baige Xu, Takaharu Yaguchi
NeurIPS 2025 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Numerical results show that our approach certainly learns energy-behavior-preserving numerical methods that are more accurate than existing numerical methods for various differential equations, including chaotic Hamiltonian systems, dissipative systems, and a nonlinear partial differential equation. We also provide universal approximation theorems for the proposed approach. Section 5 is dedicated to experiments. |
| Researcher Affiliation | Academia | Elena Celledoni Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim, Norway EMAIL; Brynjulf Owren Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim, Norway EMAIL; Chong Shen Graduate School of Science Kobe University Kobe, Japan EMAIL; Baige Xu Graduate School of Science Kobe University Kobe, Japan EMAIL; Takaharu Yaguchi Graduate School of Science Kobe University RIKEN AIP Kobe, Japan EMAIL |
| Pseudocode | No | The proposed method is formulated as follows: NNH(u(n), u(n+1)) := H(u(n), u(n+1)) + MNN(u(n), u(n+1))(u(n+1) u(n)), (9) where MNN(u(n), u(n+1)) is a skew-symmetric matrix learned by a neural network that takes u(n) and u(n+1) as the input. The universal approximation properties are the direct consequences of the above theorems. The paper describes the method using mathematical formulations and theorems but does not contain a distinct pseudocode or algorithm block. |
| Open Source Code | Yes | The implementation code is available in supplementary material on Open Review: https://openreview.net/forum?id=G2uILEbcLF. |
| Open Datasets | No | We use two sets of data: the flow data and the irregular data for the Hénon-Heiles equation. The training flow data {(u(n 1), u(n)) | n = 1, ..., N} consists of N = 100 pairs of points obtained from the trajectory calculated using a high-order integrator sampled with a time step h = 0.3, starting from u(0) = (q(0) 1 , q(0) 2 , p(0) 1 , p(0) 2 ) = (0.3, 0.3, 0.2, 0.2). The paper generates its own datasets for experiments and does not provide public access to them, nor does it use pre-existing public datasets. |
| Dataset Splits | Yes | The training flow data {(u(n 1), u(n)) | n = 1, ..., N} consists of N = 100 pairs of points obtained from the trajectory calculated using a high-order integrator sampled with a time step h = 0.3, starting from u(0) = (q(0) 1 , q(0) 2 , p(0) 1 , p(0) 2 ) = (0.3, 0.3, 0.2, 0.2). After training, we use the trained integrator to compute the flow starting at u(N) for 300 steps. We also conducted experiments using irregular data. The training dataset {((u(i 1), h(i)), u(i)) | i = 1, . . . , I} consists of I = 100 one-step input-output pairs, where the elements of u(i 1) are randomly sampled from the domain [ 1, 1], and the corresponding step sizes h(i) are randomly chosen from [0, 0.5]. Each u(i) represents the one-step solution computed from u(i 1) using step size h(i). After training, we simulate the flow starting from the initial state u(0) = (q(0) 1 , q(0) 2 , p(0) 1 , p(0) 2 ) = (0.3, 0.3, 0.2, 0.2) using a fixed step size h = 0.3 for 100 time steps. Similar details are provided for other systems. |
| Hardware Specification | Yes | All experiments were performed on an Intel i5-13500H CPU, and computations were done in double precision. In the experiment, we used an Intel Xeon 6900P CPU and an AMD Instinct MI300A GPU for computation. |
| Software Dependencies | No | We used standard multilayer perceptrons with the hyperbolic tangent (tanh) as the activation function to learn the matrix MNN. The optimizer is set to Adam [33]. As for the loss function, we chose MSELoss (Mean Squared Error Loss). We implemented the PINNs as in [54] with energy conservation/dissipation regularization terms, using the Deep XDE library, and applied it to three benchmark systems: Hénon Heiles, double pendulum, and Duffing oscillator. We also implemented the Time-Reversal Symmetric ODE Network (TRS-ODEN) [28]. The computation time for generating the training data was 28.709 sec using a CPU and the scipy odeint function. Specific version numbers for software components like Adam, Deep XDE, or SciPy are not provided. |
| Experiment Setup | Yes | The hyperparameters for training are provided in Appendix D.1. The optimizer is set to Adam [33]. As for the loss function, we chose MSELoss (Mean Squared Error Loss). Table 2: hyperparameters PROBLEM LR EPOCHS LAYER WIDTH PENDULUM -FLOW 0.0001 10000 5 50 -IRREGULAR 0.0001 10000 5 50 -DISSPATIVE 0.0001 10000 5 50 HÉNON-HEILES -FLOW 0.00001 30000 10 100 -IRREGULAR 0.00001 30000 10 100 2-BODY 0.0001 10000 5 50 2-PENDULUM 0.0001 10000 5 50 DUFFING 0.0001 10000 5 50 KDV 0.0001 10000 5 200 |