Deep-Koopman-KANDy

Introduction

The introduction to this README is plainly written by a human to detail the experimental procedures and the ideas of our method and how we came to our conclusions. We pruposley include all experiments in this repository rather

Later sections of this README are generated by an LLM that detail the code structure.

This repository contains the scientific code and worklflow for the paper:

Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics

by Kevin Slote, Erik Bollt, and Jeremie Fish Clarkson Center for Complex Systems Science Clarkson University Postdam, NY, USA

This software repository contains at a top level the code needed to reproduce and run the Deep-Koopman-KANDy algorithm

The software experiments were written with Claude Opus 2.6. The agent memory is included ad the code was verified and ran indepedenty.

We include all of the experiments that follow a narrative arc of how our algorithm was developed. The first pass of analysis included the development of the stable propogator. The code in the experiments directory with naming conventions containing stable, unitary, or ergodic, correspond to an experiment testing which contraint on the Koopman operator K, yeilded the best results.

Proceeding along the narative arc of how we arrived at our conclusions, we tested for model perormance and network sparcisty eventually arriving at our bet pykan backend model lorenz_5d_third_test.py. Then we began testing across random seeds and different ics

We then began testing against the SINDy-ae method as this was the closed model to our and given the lack of a coherent similar benchmark of dictionary extraction and we used symbolic regression to test the quality of the learned autoencoders.

We include numerous experiments detailing the optimal level-set-manifold extraction of the non-linear activations. Althought, the KANDy model was yeilding better results for some types of functions. This full analysis was mostly omitted from the paper.

The mathematics of our method were developed and put as readme files in the research_materials directory.

The code that produced the tables and figures in the paper are as follows

Run every command from the repository root. Scripts in experiments/ that are not listed below are development experiments and are not needed to reproduce the paper. Tables and figures use the numbering in the camera-ready PDF.

Models

Train these first. Each later section assumes the checkpoints below exist, and all of them are included in results/.

Model	Checkpoint	Command
Lorenz [3,5,5], unpruned	results/lorenz_5d_second_test/model.pt	python experiments/lorenz_5d_second_test.py
Lorenz [3,5,5], pruned (best model)	results/lorenz_5d_third_test/model_pruned_0.03.pt	python experiments/lorenz_5d_third_test.py (needs the unpruned model)
Lorenz [3,8,32] stable	results/lorenz_stable/stable_8_32_model.pt	python experiments/lorenz_stable.py
Standard map [32,128] stable	results/standard_map/standard_map_stable_32_128.pt	python experiments/standard_map_stable.py
Arnold cat map [4,24,1024]	results/cat_map_standard/cat_map_model.pt	python experiments/cat_map_standard.py --epochs 200
Arnold cat map, Table 6 sizes	results/table_6/model_4_*.pt	python experiments/table_6.py
Ikeda [2,4,4]	results/ikeda_minimal/model_seed_0.pt	python experiments/ikeda_minimal.py
KS (cached data and first run)	results/ks_stable/ks_data.pt, results/ks_stable/ks_stable_model.pt	python experiments/ks_stable.py
KS, wide model	results/ks_stable_wide/ks_stable_model.pt	python experiments/ks_stable.py --epochs 500 --rollout 1 --outdir results/ks_stable_wide

cat_map_standard.py uses the stable propagator, despite its name. It is cat_map_stable.py plus the Fourier and latent-decay analysis. To evaluate its saved model, pass --skip-train --epochs 200; the default of 150 epochs does not match the saved loss history and the loss plot fails.

Figures

Figure	Content	Script	Notes
1	Architecture	None	Drawn by hand: docs/figs/Network.png
2	Activation manifold	None	Edited by hand: docs/figs/manifold_publication_edited.pdf
3	Lorenz attractor and observable	experiments/fig_lorenz_paper.py	Edited by hand afterward. experiments/lorenz_paper_figures.py is a later redraw
4	Standard map phase portraits	experiments/standard_map_hero.py	Edited by hand afterward
5	Arnold cat map rollouts	experiments/cat_map_standard.py	Panels assembled by hand
6	Ikeda level sets	experiments/ikeda_levelset_analysis.py	experiments/ikeda_paper_figures.py is a redraw
7	KS spacetime rollout	experiments/ks_stable.py (wide run)	experiments/ks_paper_figures.py redraws from the saved model
8	KS NRMSE	experiments/ks_stable.py (wide run)	experiments/ks_paper_figures.py redraws from the saved model

The KS dispersion and foliation figures in the appendix also come from the wide ks_stable.py run.

Tables

collect_all_results.py assembles Tables 1 to 3 from saved results:

python experiments/collect_all_results.py --section 14  # Table 1
python experiments/collect_all_results.py --section 16  # Table 2
python experiments/collect_all_results.py --section 15  # Table 3

Sections 1 to 13 of the same script print raw numbers from the development experiments.

Table	Content	Scripts
1	Lorenz ablation and baselines	ablation_lorenz.py, ablation_lorenz_with_score.py, pykan_pruned_sweep.py, sindy_ae_only.py, edmd_dl_ablation.py, then section 14
2	Best Lorenz model summary	lorenz_5d_third_test.py, lorenz_5d_best_model_plots.py, eigenmode_dictionary.py, pykan_pruned_sweep.py, then section 16
3	Lorenz level-set decomposition	lorenz_5d_third_test.py, then section 15 (recomputed from model_pruned_0.03.pt)
4	Ikeda level-set decomposition	ikeda_minimal.py, then ikeda_levelset_analysis.py
5	Cat map error vs. horizon	table_5.py
6	Cat map model size	table_6.py
7	Diagnostic decision procedure	None (written by hand, no data)
8	Predicted vs. observed residual scales	None (predicted values are calculated by hand)
9	Lorenz Lasso sweep	pykan_pruned_sweep.py
10	Lorenz prune threshold sweep	lorenz_5d_third_test.py, pruning_stability_test.py
11 to 14	Ikeda robustness	ikeda_robustness.py
15, 16	Standard map robustness	standard_map_robustness.py
17	Prior-art benchmark	benchmark_prior_art.py

Table 5, using the saved model:

python experiments/table_5.py --checkpoint results/cat_map_standard/cat_map_model.pt

Table 6 trains any size without a checkpoint in results/table_6/ and never replaces an existing one. To train only the two small sizes and reuse the saved large model:

python experiments/table_6.py --arch 4,16,32 4,32,64
cp results/cat_map_standard/cat_map_model.pt results/table_6/model_4_24_1024.pt
python experiments/table_6.py

The horizon in Table 6 is the number of steps before angular MSE reaches 0.03. The epoch count for [4,32,64] was not recorded; table_6.py uses 150.

Table 17 needs results/ks_stable/ks_data.pt and the saved Mackey-Glass data in results/mackey_glass/. It also needs PyResDMD, which is not on PyPI.

Numbers reported in the text

Result	Script
Lorenz spectral analysis (Section 4.1)	experiments/eigenmode_dictionary.py
Standard map singular values (Section 4.2)	experiments/standard_map_contractive.py
Cat map Fourier coefficients, latent decay, dictionary (Appendix D)	experiments/cat_map_standard.py
KS dictionary	experiments/ks_lars_recheck.py, experiments/ks_lars_geometry.py
KS Fourier basis test	experiments/ks_fourier_basis_test.py
KS lagged self-predictability	experiments/ks_stable.py