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..
Hamiltonian Neural PDE Solvers through Functional Approximation
Authors: Anthony Zhou, Amir Barati Farimani
NeurIPS 2025 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We evaluate on three PDE systems and find both high accuracy and generalization capabilities, which we hypothesize arise from inductive biases to conserve energy. Code and datasets for this work are released at https://github.com/anthonyzhou-1/hamiltonian_pdes. 4 Experiments To better understand and situate integral kernel functionals, we examine their ability to model simple, analytically constructed functionals and compare their performance to other architectures. The experimental setup is as follows: given a functional F[u], random polynomials u(x) = c0xp + c1xp 1+. . .+cp 1x+cp are generated by uniformly sampling {ci [a, b] : i = 0, . . . , p}. Therefore, the dataset consists of N pairs of polynomials and evaluated functionals (un(x), F[un(x)]) for n = 1, . . . , N. In practice, each function un(x) is discretized at a set of points {xi : i = 1, . . . , M}, such that it is represented as un = [un(x1), . . . , un(x L)], and F[un] remains a scalar. |
| Researcher Affiliation | Academia | Anthony Zhou Carnegie Mellon University EMAIL Amir Barati Farimani Carnegie Mellon University EMAIL |
| Pseudocode | Yes | Algorithm 1 Training a HNS 1: repeat 2: Hθ Pn i=1 κθ(xi, ui)ui µi x 3: δHθ δu autograd(Hθ, u) 4: L = || δHθ δu ||2 or ||J ( δHθ 5: θ Update(θ, θL) 6: until converged |
| Open Source Code | Yes | Code and datasets for this work are released at https://github.com/anthonyzhou-1/hamiltonian_pdes. |
| Open Datasets | Yes | Code and datasets for this work are released at https://github.com/anthonyzhou-1/hamiltonian_pdes. |
| Dataset Splits | Yes | We restrict the degree of un(x) to p = 2 and sample ci [ 1, 1]; 100 training and 10 validation samples are generated for each case. [...] For the Advection equation, 1024/256 samples are generated for train/validation, and for the Kd V equation, 2048/256 samples are generated. [...] 256 samples are generated for training and 64 samples are generated for validation. To evaluate model generalization, we generate an additional test set of 64 samples with a random Gaussian pulse as initial conditions (Pulse). |
| Hardware Specification | Yes | Each run uses a single NVIDIA RTX 6000 Ada GPU. Numerical solvers are run on a AMD Ryzen Threadripper PRO 5975WX 32-Core CPU due to lack of GPU compatibility. |
| Software Dependencies | Yes | Data for the 2D SWE system is generated from Py Claw [30 32] using random sinusoidal initial conditions (Sines) for the height: [...] [30] Clawpack Development Team. Clawpack software, 2024. URL http://www.clawpack.org. Version 5.11.0. |
| Experiment Setup | Yes | Hyperparameters Each model has different hyperparameters for different PDEs (Adv/Kd V/SWE). We provide key hyperparameters for these experiments in Tables 10, 11, 12. Tables 10, 11, 12: FNO Parameters (Modes, Width, Layers), Unet Parameters (Width, Bottleneck Dim, Layers), HNS Parameters (Width, Kernel, Cond.) |