Learning Chaos In A Linear Way
Authors: Xiaoyuan Cheng, Yi He, Yiming Yang, Xiao Xue, Sibo Cheng, Daniel Giles, Xiaohang Tang, Yukun Hu
ICLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Our experiments on a variety of chaotic systems, including Lorenz systems, Kuramoto Sivashinsky equation and Navier Stokes equation, demonstrate that PFNN has more accurate predictions and physical statistics compared to competitive baselines including the Fourier Neural Operator and the Markov Neural Operator. 4 NUMERICAL EXPERIMENTS AND ABLATION STUDY |
| Researcher Affiliation | Academia | 1Dynamic Systems, University College London, United Kingdom 2Statistical Science, University College London, United Kingdom 3CEREA, ENPC and EDF R&D, Institut Polytechnique de Paris, France 4Computer Science, University College London, United Kingdom |
| Pseudocode | Yes | We provide the pseudocode in Appendix E. E ALGORITHM |
| Open Source Code | Yes | Code details of numerical experiments are available at https://github.com/Hy23333/PFNN. |
| Open Datasets | Yes | (1) Lorenz 63 (Lorenz, 1963): A 3-dimensional simplified model of atmospheric convection, known for its chaotic behavior and sensitivity to initial conditions. (2) Lorenz 96 (Lorenz, 1996): A surrogate model for atmospheric circulation, characterized by a chain of coupled differential equations. (3) Kuramoto-Sivashinsky equation (Papageorgiou & Smyrlis, 1991): A fourth-order nonlinear partial differential equation that models diffusive instabilities and chaotic behavior in systems, such as fluid dynamics, and reaction-diffusion processes. We sampled a 128-dimensional dataset consisting of 1,000 trajectories with 500 timesteps from the dataset 11. The description of the three dynamical systems is listed in Appendix F). 11Dataset for Kuramoto-Sivashinsky: https://zenodo.org/records/7495555 |
| Dataset Splits | Yes | Data generation: we generated 1800 trajectories for training and 200 trajectories for testing, where each trajectory contains 2000 timesteps with integration time 0.01 and a sample rate of 10. Meanwhile, in the best alignment with real scenarios, the initial conditions for trajectories in the training and testing set were drawn from a normal distribution. Dataset processing for training and testing: The dataset of Kuramoto-Sivashinsky simulations we utilized was obtained from the public source, consisting of 1200 simulated trajectories with 512 spatial dimensions, u(x, t), on the periodic boundary. ... We used 1000 trajectories for training and 200 trajectories for testing, where each trajectory was truncated to 1990 timesteps to preserve the contraction phase before the system reached the ergodic state. For model training, the full trajectory length (1990 timesteps) was used to train baseline models, while the trajectory before the contraction step k was used to train the PFNN contraction operator ˆGc, and the trajectory after k was used to train the PFNN measure-invariant operators ˆGc and ˆG m. |
| Hardware Specification | Yes | Both the training and evaluations are conducted on multiple A100s and Mac Studio with a 24-core Apple M2 Ultra CPU and 64-core Metal737 Performance Shaders (MPS) GPU. |
| Software Dependencies | Yes | The model is implemented with Py Torch version 2.3.1, and the details of the layers are presented in Table 8. |
| Experiment Setup | Yes | Training details. We use the Adam optimizer to minimize the relative L2 loss with a learning rate of 1e-4, and a step learning rate scheduler that decays by half every 10 epochs, for a total of 100 epochs. For PFNN models, when training the PFNN (consist) model, we discarded the initial 1000 timesteps to avoid the dissipative process; whilst for training the PFNN (contract) model, the initial steps were maintained to specialize the model in learning the dissipativity in the early-stage emulation. |