Robust-EPDE

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Program complex for robust training of models in the form of differential equations.

The algorithms used:

• EPDE (Evolutionary Partial Differential Equations). Algorithm for the search of differential equations based on data (https://github.com/ITMO-NSS-team/EPDE)
• BAMT (Bayesian Analytical and Modelling Toolkit). Repository of a data modeling and analysis tool based on Bayesian networks. (https://github.com/aimclub/BAMT)
• TEDEouS (Torch Exhaustive Differential Equation Solver). Algorithm for automated solution of differential equations based on neural networks (https://github.com/ITMO-NSS-team/torch_DE_solver)

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Examples & Tutorials

Before starting the program component, the configuration of each module must be set up. This is done for each task separately in the folder tasks/. The file should be named tasks/example_{title}.py. Where all additional functions are implemented, starting with def load_data():

    def load_data():
    """
        path -> data -> parameters -> derivatives (optional) -> grid -> 
        -> boundary conditions (optional) -> modules config (optional)
    """
    path = """YOUR CODE HERE"""
    data = """YOUR CODE HERE"""

    derives = None  # if there are no derivatives

    grid = """YOUR CODE HERE"""
    params = """YOUR CODE HERE"""
    
    domain = """YOUR CODE HERE"""
    boundaries = False  # if there are no boundary conditions

    noise = False
    variance_arr = ["""YOUR CODE HERE"""] if noise else [0]

    global_modules = {
        "global_config": {
            "discovery_module": "EPDE",
            "dimensionality": # (starts from 0 - [t,], 1 - [t, x], 2 - [t, x, y])
        }
    }

    epde_config = {"""YOUR CODE HERE"""}

    bamt_config = {"""YOUR CODE HERE"""}

    solver_config = {"""YOUR CODE HERE"""}

    config_modules = {**global_modules,
                      **epde_config,
                      **bamt_config,
                      **solver_config}

    with open(f'{path}config_modules.json', 'w') as fp:
        json.dump(config_modules, fp)

    cfg_ebs = config_modules.Config(f'{path}config_modules.json')

    return data, grid, derives, cfg_ebs, domain, params, boundaries

The default configuration and corresponding parameters of each module can be viewed in default_configs.py. The task instructions are then added to the main ebs_main.py file:

tasks = {
    'wave_equation': example_wave_equation
}

title = list(tasks.keys())[0] # name of the problem/equation

The data archive for equation discovery and recovery is here.

Results

1. Wave equation with one spatial variable (more)

$$\frac{\partial^{2} u}{\partial t^{2}} - \frac{1}{25} \frac{\partial^{2} u}{\partial x^{2}} = 0,$$ $$\\ 100\times100, x \in [0; 1], t \in [0; 1].$$

The output of the EPDE module is presented in the form of a table. The fields in the table are the structures of the obtained partial differential equations, where each row contains the coefficients at each structure.

These data are input to the BAMT module to build a Bayesian network based on them. The results of the module are:

• distribution of coefficients at structures (illustrated in the figure below)
• list of sampled partial differential equations, which in turn is the input to the SOLVER module.

The resulting solution fields of partial differential equations are used to construct a confidence region and an average solution. The results are displayed for comparison with the original data and as heat maps.

2. Burgers' equation

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,$$ $$\\ 256\times256, x \in [-4000; 4000], t \in [0; 4].$$

3. Burgers' equation (reduced grid)

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0,$$ $$\\ 100 \times100, x \in [-1000; 0], t \in [0; 1].$$

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