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Project 3: Foundation Models for Phase-Field Dynamics

In this project, we aim to train and evaluate a foundation model 1. Our implementation is based on the Fourier Neural Operator (FNO) 2 as its architectural backbone.

Within the scope of our project, we consider the 1D Allen-Cahn equation with varying $\epsilon$ values:

$$ \frac{\partial u}{\partial t} = \Delta_{x} u - \frac{1}{\epsilon^{2}} (u^{3} - u), x\in [-1,1] $$

This README.md serves only as a brief overview on how to run our code. For more details, please see the project 3 section of the main project report, where we explained the selection of our model architecture and capabilities & limitations of our model. We also provided a proof for the error estimate of Allen-Cahn equation, which was a bonus task for the project.

Getting Started

Note

  • The prediction file results/prediction.txt is the redirection of the I/O by running python3 evaluate.py > results/prediction.txt.

Data Generation

For data generation, we need to first generate initial conditions (IC) $u_{0}= u(x, 0)$, and then solve the system with scipy.integrate.solve_ivp. To generate various types of initial conditions, we implemented three different samplers of the FunctionSampler class that provides three types of IC:

  1. Piecewise Linear (PL)
  2. Gaussian Mixture (GM)
  3. Fourier Series (FS)

The generated datasets train_sol.npy, test_sol.npy, test_sol_OOD.npy and test_sol_eps.npy are already included under data/ folder. If one wish to reproduce the data generation, either run without flags:

python3 data_generator.py

Or one could also specify to just plot (--plot) or generate data (--generate), by passing flags respectively.

In the following table, one can see for each dataset 5 selected samples at initial time $t = 0$. The first two columns consist of in-distribution initial conditions, where we used samplers with default parameters. On the contrary, the OOD dataset test_sol_OOD.npy exhibits very distinct nature. We specified entirely different parameters at the initialization of each sampler, to let the sampled p.w. linear function to have more breakpoints, Gaussian mixture to include more components and the Fourier series to sum up to higher index for generating data of higher frequency pattern and sharper transitions. All the parameters used for data generation can be found under config.json for each data folder.

To test out the capability of our foundation model to extrapolate aka. interpolate for different $\epsilon$ values, we test it on the dedicated dataset test_sol_eps.npy. This dataset investigate different epsilon values, and use default samplers as training dataset.

Training Data (train_sol.npy) Testing Data (test_sol.npy) OOD Testing Data (test_sol_OOD.npy)
training testing ood testing

Solution Trajectories Across 5 Timestamps

In total, a fixed number of timestamps nt = 5 are used. The temporal grid is discretized uniformly, and thus all timesteps are of size $\Delta t = 0.0025$.

Training Data (train_sol.npy) Testing Data (test_sol.npy) OOD Testing Data (test_sol_OOD.npy)
In-distribution EvolutionEvolution of Selected trajectories at different ɛ from train_sol.npy (In-distribution) Test EvolutionEvolution of Selected trajectories at different ɛ from test_sol.npy (In-distribution) OOD EvolutionEvolution of Selected trajectories at different ɛ from test_sol_OOD.npy (OOD)
OOD Testing Data (test_sol_eps.npy)
OOD EvolutionEvolution of Selected trajectories at different ɛ from test_sol_eps.npy (OOD)

Training

To train the base model, run:

python3 training.py

The trained base model will be stored under checkpoints/ace_* with current timestamp at running under the name base_model.pth. One could optionally also train our model with curriculum learning technique by passing --curriculum flag.

Evaluation

One can use our provided pretrained model under checkpoints/ace_fno_m20_w64_d2_20250116_215544 for direct evaluation by running:

python3 evaluate.py

This will use the fine-tuned models for each dataset as comparison in the evaluation.

For fine-tuning, we used 20 trajectories (with 80/20 split for training vs. validation) for each dataset and set the epochs to 100. The base_model.pth is loaded to serve as the starting point and we freeze the input & output layers, as well as the FiLM layers to focus fine-tuning only on the FNO blocks.

Results

Following, we report our results on all datasets across time snapshots. In each single figure, the solution evolves from the identical initial condition $u(x, 0)$, but with different $\epsilon$ parameter value used. We also report the error percentage, comparing results from fine-tuned model and our base model.

These results are obtained from default training technique taking into account of all input data types at once, without curriculum update.

test_sol.npy

Piecewise Linear (PL) Gaussian Mixture (GM) Fourier Series (FS)
pl gm fs

test_sol_OOD.npy

Piecewise Linear (PL) Gaussian Mixture (GM) Fourier Series (FS)
pl gm fs

test_sol_eps.npy

Piecewise Linear (PL) Gaussian Mixture (GM) Fourier Series (FS)
pl gm fs

Footnotes

  1. Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior S. Subramanian et al. Advances in Neural Information Processing Systems, vol. 36, 2023 [paper] ↩

  2. Fourier Neural Operator for Parametric Partial Differential Equations
    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar
    International Conference on Learning Representations (ICLR)
    [paper] [blog] ↩