dataset.py

"""
2D Poisson equation dataset with mixed boundary conditions.

Solves the boundary value problem:
    ∇²u = f(x,y)     in Ω = [0, Lx] × [0, Ly]
    u(x,0) = g(x)    on bottom boundary  
    ∂u/∂y(x,Ly) = h(x) on top boundary

Uses Dedalus spectral methods with Fourier(x) × Chebyshev(y) discretization.
The forcing function f(x,y) is randomly generated using Gaussian processes
to create diverse training samples.

Physical applications: electrostatic potential, steady-state heat conduction,
fluid stream functions.
"""

import numpy as np
from torch.utils.data import IterableDataset

import dedalus.public as d3
from functools import partial
from sklearn.metrics.pairwise import rbf_kernel
import logging

logger = logging.getLogger(__name__)


def sample_gp_prior(kernel, X, n_samples=1):
    """
    Sample from Gaussian Process prior for generating smooth random fields.
    """
    if X.ndim == 1:
        X = X.reshape(-1, 1)

    K = kernel(X, X)

    prior = np.random.multivariate_normal(
        mean=np.zeros(X.shape[0]),
        cov=K,
        size=n_samples,
    )

    return prior


def sample_gp_posterior(kernel, X, y, xt, n_samples=1):
    """Sample from Gaussian Process posterior for boundary conditions."""
    if X.ndim == 1:
        X = X.reshape(-1, 1)
    if y.ndim == 1:
        y = y.reshape(-1, 1)
    if xt.ndim == 1:
        xt = xt.reshape(-1, 1)

    K = kernel(X, X)
    Kt = kernel(X, xt)
    Ktt = kernel(xt, xt)

    K_inv = np.linalg.inv(K)

    mu = Kt.T @ K_inv @ y
    cov = Ktt - Kt.T @ K_inv @ Kt

    mu = mu.ravel()  # Ensure 1D array

    posterior = np.random.multivariate_normal(
        mean=mu,
        cov=cov,
        size=n_samples,
    )
    return posterior


class PoissonDataset(IterableDataset):
    """
    Dataset for 2D Poisson equation with mixed boundary conditions.
    
    Generates random smooth forcing functions using Gaussian processes
    and solves: ∇²u = f(x,y) with u(x,0) = g(x) and ∂u/∂y(x,Ly) = h(x).
    """
    def __init__(
        self,
        Lx=2*np.pi,              # Domain length in x-direction
        Ly=np.pi,                # Domain length in y-direction  
        Nx=256,                  # Grid points in x-direction
        Ny=128,                  # Grid points in y-direction
        dtype=np.float64,
    ):
        """
        Initialize 2D Poisson equation dataset.
        
        Args:
            Lx: Domain length in x-direction
            Ly: Domain length in y-direction  
            Nx: Number of grid points in x-direction
            Ny: Number of grid points in y-direction
            dtype: Data type for computations
        """
        super().__init__()
        
        # Store domain and grid parameters
        self.Lx = Lx
        self.Ly = Ly
        self.Nx = Nx  
        self.Ny = Ny
        self.dtype = dtype
        
        # Setup Dedalus solver components
        self.coords = d3.CartesianCoordinates('x', 'y')
        self.dist = d3.Distributor(self.coords, dtype=dtype)
        self.xbasis = d3.RealFourier(self.coords['x'], size=Nx, bounds=(0, Lx))
        self.ybasis = d3.Chebyshev(self.coords['y'], size=Ny, bounds=(0, Ly))
        
        # Create coordinate grids
        self.x, self.y = self.dist.local_grids(self.xbasis, self.ybasis)
        
        # Setup fields
        self.u = self.dist.Field(name='u', bases=(self.xbasis, self.ybasis))
        self.tau_1 = self.dist.Field(name='tau_1', bases=self.xbasis)
        self.tau_2 = self.dist.Field(name='tau_2', bases=self.xbasis)
        self.f = self.dist.Field(bases=(self.xbasis, self.ybasis))
        self.g = self.dist.Field(bases=self.xbasis)
        self.h = self.dist.Field(bases=self.xbasis)
        
        # Setup operators and namespace for boundary value problem
        dy = lambda A: d3.Differentiate(A, self.coords['y'])
        lift_basis = self.ybasis.derivative_basis(2)
        lift = lambda A, n: d3.Lift(A, lift_basis, n)
        f = self.f
        g = self.g  
        h = self.h
        u = self.u
        tau_1 = self.tau_1
        tau_2 = self.tau_2
        Ly = self.Ly
        
        self.problem = d3.LBVP([self.u, self.tau_1, self.tau_2], namespace=locals())
        self.problem.add_equation("lap(u) + lift(tau_1,-1) + lift(tau_2,-2) = f")
        self.problem.add_equation("u(y=0) = g")
        self.problem.add_equation("dy(u)(y=Ly) = h")

    def __iter__(self):
        """Generate infinite samples from the dataset."""
        while True:
            # Generate random forcing function using filtered noise
            self.f.fill_random('g', seed=np.random.randint(0, 2**31))
            filter_x = np.random.randint(32, 128)
            filter_y = np.random.randint(16, 64)
            self.f.low_pass_filter(shape=(filter_x, filter_y))
            
            # Generate random Dirichlet boundary condition using GP
            # Get x-coordinates for bottom boundary (1D array)
            x_boundary = self.dist.local_grid(self.xbasis)
            sigma_g = 0.2 * self.Lx
            gamma_g = 1 / (2 * sigma_g**2)
            g_sample = sample_gp_posterior(
                kernel=partial(rbf_kernel, gamma=gamma_g),
                X=np.array([0, self.Lx]),
                y=np.array([0, 0]),
                xt=x_boundary.ravel(),
                n_samples=1
            )[0]
            self.g['g'] = (np.random.uniform(0.01, 0.1) * g_sample).reshape(-1, 1)
            
            # Generate random Neumann boundary condition using GP  
            sigma_h = 0.3 * self.Lx
            gamma_h = 1 / (2 * sigma_h**2)
            h_sample = sample_gp_posterior(
                kernel=partial(rbf_kernel, gamma=gamma_h),
                X=np.array([0, self.Lx]),
                y=np.array([0, 0]),
                xt=x_boundary.ravel(),
                n_samples=1
            )[0]
            self.h['g'] = (np.random.uniform(0.005, 0.05) * h_sample).reshape(-1, 1)
            
            yield self.solve()

    def solve(self):
        """
        Solve the 2D Poisson boundary value problem.
        
        The forcing function and boundary conditions have already been set in __iter__.
        
        Returns:
            A dictionary containing the solution and related data.
        """
        
        # Build and solve the LBVP
        solver = self.problem.build_solver()
        solver.solve()
        
        # Gather global data for return
        x_global = self.xbasis.global_grid(self.dist, scale=1)
        y_global = self.ybasis.global_grid(self.dist, scale=1) 
        u_global = self.u.allgather_data('g')
        f_global = self.f.allgather_data('g')
        g_global = self.g.allgather_data('g')
        h_global = self.h.allgather_data('g')
        
        # Create coordinate meshgrids
        X, Y = np.meshgrid(x_global.ravel(), y_global.ravel(), indexing='ij')

        return {
            # Combined spatial coordinates
            "spatial_coordinates": np.array([X, Y]),  # Shape: (2, Nx, Ny)
            
            # Solution field
            "solution_field": u_global,  # Shape: (Nx, Ny)
            
            # Forcing function 
            "forcing_function": f_global,  # Shape: (Nx, Ny)
            
            # Boundary conditions
            "boundary_condition_bottom": g_global,  # Shape: (Nx,)
            "boundary_condition_top_gradient": h_global,  # Shape: (Nx,)
        }