Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Symbolic Neural Ordinary Differential Equations
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 Symbolic Neural ODEs (SNODEs), a three-stage framework that pre-trains a Fourier-based symbolic network via gradient flow matching, fine-tunes it with Neural ODEs, and adds a general neural network for residuals to learn parametric ODE/PDE operators.
Problem
Learning operator mappings from parameter functions to the states of parametric ODE/PDE systems in an interpretable way that extrapolates beyond training distributions and is resolution-invariant.
Approach
Combines a continuous-depth symbolic network (SymNet) whose first layer contains Fourier-computed spatial derivative terms with a general neural network (GeNN) for residuals, trained in three stages (gradient flow matching pre-training, ODESolve fine-tuning, residual learning) with a universal approximation result.
Key results
- On the DR, KS, NS tables with N_tr=1000 and no noise, SNODEs achieves MSE of about 3e-6, 2e-5, and 6e-4 versus 0.67, 2.56, and 7.30 for DeepONet.
- With 100 training trajectories and no noise, SNODEs reaches MSE 0.11 on NS versus 6.65 for MP-PDE and 43.9 for FNO.
- Training time for SNODEs is 221 s (DR), 489 s (KS), and 824 s (NS), versus 1580, 2920, and 12476 s for NODE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SNODEs instead of the strongest listed baseline | Parametric PDE modeling · DR, KS, and NS under all reported training sizes and noise levels | MSE | Helped |
| Remove SymNet (ablation) | SNODEs · DR, KS, and NS, N_tr=100 and N_tr=1000 with σ_n=0% | MSE | Hurt |
| Remove SymNet (ablation) | SNODEs · DR, KS, and NS, N_tr=1000 with σ_n=3% | MSE | Mixed |
| Remove GeNN (ablation) | SNODEs · DR, KS, and NS, N_tr=100 and N_tr=1000 with σ_n=0% | MSE | Mixed |
| Remove GeNN (ablation) | SNODEs · DR, KS, and NS, N_tr=1000 with σ_n=3% | MSE | Mixed |
| Use flow-matching pre-training rather than directly training with the classic NODE method | SNODEs training · DR, KS, and NS | training time | Helped |
Surprising findings
- Under 3% noise with 1000 trajectories, SNODEs' MSE rises to 0.53 (DR), 0.41 (KS), and 1.37 (NS), so noise degrades it substantially.
Limitations
- Fourier transform is suited mainly to uniformly spaced spatial data; complex geometries need non-uniform transforms or other methods.
- Deep SymNet may become unstable when underlying dynamics are complex.
- Performance degrades significantly under noisy training data (e.g., 3% noise).
Experimental setup
Datasets: simulated DR, KS, and 2-d NS trajectories from pseudospectral solver simulated 1-d ODE and bifurcation trajectories with GRF parameter functions
Benchmarks: 1-d parametric ODE (DeepONet nonlinear example) saddle-node, pitchfork, Hopf bifurcation systems diffusion-reaction (DR) Kuramoto-Sivashinsky (KS) 2-d Navier-Stokes (NS)
Metrics: mean squared error training time (seconds)
Baselines: DeepONet FNO PDE-NET MP-PDE NODE
Builds on: Neural ODEs Symbolic Neural Network (SymNet) DeepONet Fourier Neural Operator PDE-Net
Closest papers by meaning
- From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems (ICML 2024 · similarity 0.90)
- ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence (NeurIPS 2024 · similarity 0.88)
- Closed-form Solutions: A New Perspective on Solving Differential Equations (ICML 2025 · similarity 0.88)
- Semi-supervised Learning of Dynamical Systems with Neural Ordinary Differential Equations: A Teacher-Student Model Approach (AAAI 2024 · similarity 0.87)
- Semi-Implicit Neural Ordinary Differential Equations (AAAI 2025 · similarity 0.87)
- How Deep Do We Need: Accelerating Training and Inference of Neural ODEs via Control Perspective (ICML 2024 · similarity 0.87)
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.