An effective plug-and-play scaling module for solving stiff ODEs
- [2025.12.01] GBCT code is now open-source!
We have open-sourced the full datasets, pre-trained weights (checkpoints), and training/inference scripts used in the paper. (coming soon)
| Case | Dataset | Checkpoint |
|---|---|---|
| Chemical Reaction | Training Set | Test Set | Download |
| Nuclear Reaction | Dataset | Download |
| Robertson Problem | Dataset | Download |
We propose the Generalized Box–Cox Transformation (GBCT), a nonlinear scaling method that alleviates multiscale stiffness by compressing multi-magnitude data. Integrated into our data-driven framework, DeePODE, GBCT shows improved performance across diverse stiff ODE and PDE benchmarks.
- Plug-and-Play: Can be easily integrated into deep learning-based frameworks.
- 6x Faster Training: Achieves comparable performance to baselines with only ~1/6 of the training epochs.
- Long-Term Stability: Effectively suppresses error accumulation in long-term time integration (up to 50,000 steps).
- Versatility: Validated on chemical kinetics, nuclear reactions, and the corresponding reaction-diffusion benchmarks.
We evaluated GBCTNet across six representative benchmarks, ranging from 0D stiff ODEs to 2D reacting flows.
In the DRM19 (Methane reaction kinetics) and 13-isotope nuclear networks, GBCTNet significantly outperforms the BCTNet baseline.
- Error Reduction: Relative error for equilibrium temperature reduced from ~50% (Baseline) to 0.1% (GBCTNet).
- Stability: Maintains trajectory stability over 50,000 time steps.
Figure: Long-term temperature evolution comparison.
In a 2D turbulent methane/air ignition case, GBCTNet accurately captures the flame structure and intermediate radicals.
- Radical Prediction: The maximum relative error for unstable radicals (e.g., H) is reduced from 274% (Baseline) to 100% (GBCTNet).
-
Temperature Accuracy: Relative error at
$t=2ms$ is 4.8% vs 17.8% for the baseline.
Figure: 2D Turbulent flame snapshots.
In simulating a white dwarf's internal deflagration (nuclear flame):
- Wave Speed: GBCTNet accurately predicts the flame propagation velocity (~200 m/s) and ignition timing.
- Morphology: In the 2D wedge-shaped flame test, GBCTNet captures the sharp flame front with errors confined to a narrow band.
Figure: 1D unsteady nuclear flame propagation.
GBCTNet converges significantly faster. It achieves low generalization error (RMSE) in the early training stages, requiring only 1/6 of the epochs needed by the baseline to reach comparable accuracy.
Figure: Training efficiency.
conda install pytorch
conda install --channel cantera cantera==2.6.0 -y
conda install numpy matplotlib seaborn scikit-learn pandas -y
pip install easydict scienceplots meshio -i https://pypi.tuna.tsinghua.edu.cn/simple
conda install -c conda-forge mpi4py openmpidocker pull ckode/deepck:1.0.0_pytorch1.12_cuda11.3