Part D: pySDC and PyTorch

PyTorch is a library for machine learning. Its data structure is called a tensor, which runs on CPUs as well as GPUs, with access to all of PyTorch’s machine learning methods. Since the potential for use in pySDC is very large, we have started on a data type that uses PyTorch tensors throughout pySDC.

This example trains a network to predict the results of implicit Euler solves for the heat equation. It is too simple to do anything useful, but demonstrates how to use tensors in pySDC and then apply the enormous PyTorch infrastructure. This is work in progress in very early stages! The tensor data type is the simplest possible implementation, rather than an efficient one. If you want to work on this, your input is appreciated!

The tensor data type, the problem class Heat1DFDTensor using it and the network HeatEquationModel live in the helper module torch_heat.py next to this tutorial.

from pathlib import Path

import numpy as np
import torch
import torch.nn as nn
import torch.optim as optim
from pySDC.tutorial.step_7.torch_heat import Heat1DFDTensor, HeatEquationModel

Training at the collocation nodes

For a first proof of concept, we train the model specifically for the collocation nodes we use in SDC. If successful, the initial guess would already be the exact solution, and we would need no SDC iterations.

What we find is that we can train the network to predict the solution to one very specific problem rather well. However, if we train for something else, here solving for a different step size, the model only predicts what it was trained for last, and loses the ability to solve for what it learned before. This is solely because we chose an overly simple model that is unsuitable to the task at hand, which is likely easily solved with a bit of patience. So take this as a demonstration of the interface between pySDC and PyTorch, and if you want to do a project with it, as a starting point for things that actually do something!

The output shows the training loss during training and, after each of three training sessions, the error of the prediction with the current state of the network. To demonstrate the forgetfulness, we finally print the error of all learned predictions after training is complete.

def train_at_collocation_nodes():
    """
    For the first proof of concept, we want to train the model specifically to the collocation nodes we use in SDC.
    If successful, the initial guess would already be the exact solution and we would need no SDC iterations.

    What we find is that we can train the network to predict the solution to one very specific problem rather well.
    See the error during training for what happens when we ask the network to solve for exactly what it just trained.
    However, if we train for something else, i.e. solving to a different step size in this case, we can only use the
    model to predict the solution of what it's been trained for last and it loses the ability to solve for previously
    learned things. This is solely because we chose an overly simple model that is unsuitable to the task at hand and
    is likely easily solved with a bit of patience. This is just a demonstration of the interface between pySDC and
    PyTorch. If you want to do a project with this, feel free to take this as a starting point and do things that
    actually do something!

    The output shows the training loss during training and, after each of three training sessions is complete, the error
    of the prediction with the current state of the network. To demonstrate the forgetfulness, we finally print the
    error of all learned predictions after training is complete.
    """
    out = ''
    errors_mid_training = []
    errors_post_training = []

    # instantiate the pySDC problem and a model for PyTorch
    problem = Heat1DFDTensor()
    model = HeatEquationModel(problem)

    # setup neural network
    lr = 0.001
    num_epochs = 250
    criterion = nn.MSELoss()
    optimizer = optim.Adam(model.parameters(), lr=lr)

    # setup initial conditions
    t = 0
    initial_condition = problem.u_exact(t)

    # train the model to predict the solution at certain collocation nodes
    collocation_nodes = np.array([0.15505102572168285, 0.6449489742783183, 1]) * 1e-2
    for dt in collocation_nodes:

        # get target condition from implicit Euler step
        target_condition = problem.solve_system(initial_condition, dt, initial_condition, t)

        # do the training
        for epoch in range(num_epochs):
            predicted_state = model(initial_condition, t, dt)
            loss = criterion(predicted_state.float(), target_condition.float())

            optimizer.zero_grad()
            loss.backward()
            optimizer.step()

            if (epoch + 1) % 50 == 0:
                out += f'Training for {dt=:.2e}: Epoch [{epoch+1:4d}/{num_epochs:4d}], Loss: {loss.item():.4e}\n'

        # evaluate model to compute error
        model_prediction = model(initial_condition, t, dt)
        errors_mid_training += [abs(target_condition - model_prediction)]
        out += f'Error of prediction at {dt:.2e} during training: {abs(target_condition-model_prediction):.2e}\n'

    # compare model and problem
    for dt in collocation_nodes:
        target_condition = problem.solve_system(initial_condition, dt, initial_condition, t)
        model_prediction = model(initial_condition, t, dt)
        errors_post_training += [abs(target_condition - model_prediction)]
        out += f'Error of prediction at {dt:.2e} after training: {abs(target_condition-model_prediction):.2e}\n'

    print(out)
    Path('data').mkdir(parents=True, exist_ok=True)
    with open('data/step_7_D_out.txt', 'w') as file:
        file.write(out)

    # test that the training went as expected
    assert np.greater([1e-2, 1e-4, 1e-5], errors_mid_training).all(), 'Errors during training are larger than expected'
    assert np.greater([1e0, 1e0, 1e-5], errors_post_training).all(), 'Errors after training are larger than expected'

    # save the model to use it throughout pySDC
    torch.save(model.state_dict(), 'data/heat_equation_model.pth')


if __name__ == '__main__':
    train_at_collocation_nodes()

Results

PyTorch does not run in the browser, nor in the environment this website is built in. These are the results of our CI, which runs this part in an environment with PyTorch, in the run that built this page:

Training for dt=1.55e-03: Epoch [  50/ 250], Loss: 1.0780e-02
Training for dt=1.55e-03: Epoch [ 100/ 250], Loss: 1.2480e-04
Training for dt=1.55e-03: Epoch [ 150/ 250], Loss: 1.6670e-07
Training for dt=1.55e-03: Epoch [ 200/ 250], Loss: 3.1713e-09
Training for dt=1.55e-03: Epoch [ 250/ 250], Loss: 1.5126e-11
Error of prediction at 1.55e-03 during training: 4.99e-05
Training for dt=6.45e-03: Epoch [  50/ 250], Loss: 1.5263e-04
Training for dt=6.45e-03: Epoch [ 100/ 250], Loss: 8.2049e-07
Training for dt=6.45e-03: Epoch [ 150/ 250], Loss: 3.7705e-09
Training for dt=6.45e-03: Epoch [ 200/ 250], Loss: 2.1631e-11
Training for dt=6.45e-03: Epoch [ 250/ 250], Loss: 6.6165e-14
Error of prediction at 6.45e-03 during training: 9.36e-07
Training for dt=1.00e-02: Epoch [  50/ 250], Loss: 1.6436e-05
Training for dt=1.00e-02: Epoch [ 100/ 250], Loss: 4.9155e-08
Training for dt=1.00e-02: Epoch [ 150/ 250], Loss: 3.4089e-10
Training for dt=1.00e-02: Epoch [ 200/ 250], Loss: 3.6608e-12
Training for dt=1.00e-02: Epoch [ 250/ 250], Loss: 1.5962e-14
Error of prediction at 1.00e-02 during training: 3.33e-07
Error of prediction at 1.55e-03 after training: 4.28e-01
Error of prediction at 6.45e-03 after training: 1.13e-01
Error of prediction at 1.00e-02 after training: 3.33e-07