Pinn Training

PINN Training (DeepXDE)

Overview

Solve PDEs using Physics-Informed Neural Networks. A neural network approximates the solution u(x,t), trained by minimizing the PDE residual + boundary/initial condition losses. Works for forward problems (solve PDE) and inverse problems (discover parameters from data).

When to Use

• Complex geometries where meshing is difficult
• Inverse problems (discovering PDE parameters from data)
• Noisy or sparse data with known governing equations
• Multi-physics problems with coupled PDEs
• When you need a differentiable surrogate of the PDE solution

Do NOT Use When

• Simple 1D/2D problems on regular grids (use finite differences — faster)
• You need guaranteed error bounds (PINNs don't provide rigorous bounds)
• Very high accuracy is needed (< 1e-6 relative error is hard for PINNs)

Installation

pip install deepxde
# DeepXDE auto-detects backend: TensorFlow, PyTorch, or JAX
# Set backend: export DDE_BACKEND=pytorch

Core Workflows

1. Forward Problem: 1D Heat Equation

$$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$

import deepxde as dde
import numpy as np

alpha = 0.01  # thermal diffusivity

def pde(x, u):
    """Heat equation residual: u_t - alpha * u_xx = 0"""
    du_t = dde.grad.jacobian(u, x, i=0, j=1)   # du/dt
    du_xx = dde.grad.hessian(u, x, i=0, j=0)    # d²u/dx²
    return du_t - alpha * du_xx

# Domain: x in [0, 1], t in [0, 1]
geom = dde.geometry.Interval(0, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

# Boundary conditions: u(0,t) = u(1,t) = 0
bc = dde.icbc.DirichletBC(geomtime, lambda x: 0,
                           lambda x, on_boundary: on_boundary)

# Initial condition: u(x,0) = sin(pi*x)
ic = dde.icbc.IC(geomtime, lambda x: np.sin(np.pi * x[:, 0:1]),
                  lambda x, on_initial: on_initial)

# Collocation points
data = dde.data.TimePDE(geomtime, pde, [bc, ic],
                         num_domain=2000, num_boundary=100,
                         num_initial=100, num_test=500)

# Neural network: 2 inputs (x,t) → 1 output (u)
net = dde.nn.FNN([2] + [64]*3 + [1], "tanh", "Glorot uniform")

model = dde.Model(data, net)

# Train: Adam first, then L-BFGS for fine-tuning
model.compile("adam", lr=1e-3)
losshistory, train_state = model.train(epochs=10000, display_every=2000)

model.compile("L-BFGS")
losshistory, train_state = model.train()

# Evaluate
x_test = np.linspace(0, 1, 100)
t_test = np.array([0.1, 0.3, 0.5, 0.8])
for t_val in t_test:
    X = np.column_stack([x_test, np.full_like(x_test, t_val)])
    u_pred = model.predict(X)
    u_exact = np.sin(np.pi * x_test) * np.exp(-alpha * np.pi**2 * t_val)
    error = np.max(np.abs(u_pred.flatten() - u_exact))
    print(f"t={t_val:.1f}: max error = {error:.4e}")

2. Inverse Problem: Discover Diffusion Coefficient

# Given noisy measurements of u(x,t), find alpha
alpha_var = dde.Variable(0.05)  # initial guess, will be optimized

def pde_inverse(x, u):
    du_t = dde.grad.jacobian(u, x, i=0, j=1)
    du_xx = dde.grad.hessian(u, x, i=0, j=0)
    return du_t - alpha_var * du_xx

# Add observation data
def generate_data(n_obs=50):
    x_obs = np.random.rand(n_obs, 1)
    t_obs = np.random.rand(n_obs, 1) * 0.5
    u_obs = np.sin(np.pi * x_obs) * np.exp(-0.01 * np.pi**2 * t_obs)
    u_obs += 0.01 * np.random.randn(n_obs, 1)  # noise
    return np.hstack([x_obs, t_obs]), u_obs

observe_x, observe_u = generate_data()
observe_bc = dde.icbc.PointSetBC(observe_x, observe_u)

data = dde.data.TimePDE(geomtime, pde_inverse, [bc, ic, observe_bc],
                         num_domain=2000, num_boundary=100,
                         num_initial=100)

net = dde.nn.FNN([2] + [64]*3 + [1], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=1e-3, external_trainable_variables=[alpha_var])
model.train(epochs=20000, display_every=5000)
model.compile("L-BFGS", external_trainable_variables=[alpha_var])
model.train()

print(f"Discovered alpha = {alpha_var.numpy():.6f} (true: 0.01)")

3. 2D Poisson Equation

def pde_poisson(x, u):
    du_xx = dde.grad.hessian(u, x, i=0, j=0)
    du_yy = dde.grad.hessian(u, x, i=1, j=1)
    # Source term: -2pi^2 sin(pi*x)*sin(pi*y)
    f = -2 * np.pi**2 * np.sin(np.pi * x[:, 0:1]) * np.sin(np.pi * x[:, 1:2])
    return du_xx + du_yy - f

geom = dde.geometry.Rectangle([0, 0], [1, 1])
bc = dde.icbc.DirichletBC(geom, lambda x: 0, lambda x, on_boundary: on_boundary)

data = dde.data.PDE(geom, pde_poisson, [bc], num_domain=2000, num_boundary=200)
net = dde.nn.FNN([2] + [64]*4 + [1], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
model.train(epochs=15000)
model.compile("L-BFGS")
model.train()

Training Strategy

Network Architecture Guide

Common Pitfalls