Research map / Probabilistic and scientific modeling / Differential equations and PDEs
TRENDy: Temporal Regression of Effective Nonlinear Dynamics
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes TRENDy, an equation-free method that maps PDE data through scattering-transform features and fits a parameter-conditioned neural ODE to learn low-dimensional effective dynamics, then uses it to locate Turing and Hopf bifurcations and analyze ocellated lizard skin patterning.
Problem
Learning low-dimensional, predictive, parameter-dependent effective dynamics of spatiotemporal PDEs from limited, noisy data in order to detect bifurcations in unseen parameter regions, when governing equations are unknown.
Approach
Compress PDE states with a multiscale scattering transform (top-d coefficients) into effective trajectories, then regress a neural ODE conditioned on the true PDE parameters, initialized at the effective initial state, so the learned effective dynamics can be continued in parameter space to find bifurcations without a decoder.
Key results
- On Gray Scott, TRENDy with 10 scattering coefficients estimated k*=.0624 (true .062), outperforming SINDyCP (.0651) on clean data, and degraded more gracefully under boundary and patch noise.
- On the ocellated lizard, a 4-way... quadrant SVM on TRENDy's final 10-d state reached 95% test accuracy (weighted F1 .95).
- On the Brusselator, TRENDy located the Hopf manifold qualitatively correctly in clean and low-holdout conditions, with error rising under noise and larger holdout (test RMSE from 4.34e-4 to 1.05e-2).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TRENDy with 10 scattering coefficients instead of fewer coefficients (ablation) | TRENDy on Gray Scott · Gray Scott pattern classification test | F1 score | Helped |
| Use spatial-gradient features instead of TRENDy T10 scattering features (ablation) | TRENDy on Gray Scott · Gray Scott pattern classification test | F1 score | Hurt |
| Add more scattering coefficients (ablation) | TRENDy on Gray Scott · Gray Scott bifurcation detection | bifurcation prediction performance | Helped |
| Use patches noise rather than no noise | TRENDy · Gray Scott test | RMSE: 4.64\times 10^{-1} -> 1.01\times 10^{0} | Hurt |
| Use boundaries noise rather than no noise | TRENDy · Gray Scott test | RMSE: 4.64\times 10^{-1} -> 7.59\times 10^{-1} | Hurt |
| Use patches noise rather than no noise | TRENDy · Brusselator test, epsilon=1.0 | forecasting error: 1.01\times 10^{-3} -> 1.05\times 10^{-2} | Hurt |
| Increase the holdout factor from epsilon=.015 to epsilon=1.0 | TRENDy · Brusselator test, no noise | forecasting error: 4.34\times 10^{-4} -> 1.01\times 10^{-3} | Hurt |
| Use TRENDy rather than SINDyCP | Gray Scott bifurcation detection · Gray Scott, noiseless | estimated critical k: .0624 vs .0651 | Mixed |
| Train with noise rather than without noise | TRENDy · Brusselator bifurcation manifold | manifold estimation accuracy | Hurt |
| Use TRENDy-predicted effective states to classify lizard patch quadrants | TRENDy + SVM · Ocellated lizard quadrant test | accuracy: .95 accuracy | Helped |
Limitations
- Scattering coefficients are lossy and not learned, so reconstruction is underdetermined and the feature choice is fixed.
- Current version performs poorly on high-frequency oscillatory real data due to conflicting learning signals.
- The lizard analysis is described as a proof of concept on a single animal.
Experimental setup
Datasets: Gray Scott simulations (synthetic) Brusselator simulations (synthetic) ocellated lizard high-resolution video Ising model simulations (MCMC)
Benchmarks: Gray Scott Turing bifurcation detection (k* estimate) Gray Scott 4-way pattern classification (F1) Brusselator forecasting RMSE and Hopf manifold estimation ocellated lizard quadrant classification
Metrics: estimated critical k* F1 score RMSE / MSE classification accuracy
Baselines: SINDyCP spatial gradient features Fourier vector features
Builds on: neural ODE scattering transform SINDyCP equation-free framework
Closest papers by meaning
- Let's do the time-warp-attend: Learning topological invariants of dynamical systems (ICLR 2024 · similarity 0.89)
- Neural Persistence Dynamics (NeurIPS 2024 · similarity 0.87)
- Sparse Diffusion Autoencoder for Test-time Adapting Prediction of Complex Systems (NeurIPS 2025 · similarity 0.86)
- Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems (NeurIPS 2024 · similarity 0.86)
- Curly Flow Matching for Learning Non-gradient Field Dynamics (NeurIPS 2025 · similarity 0.85)
- Identifying latent state transitions in non-linear dynamical systems (ICLR 2025 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.