heat.resample.rst

Heat equation with training points resampling

Problem setup

We will solve a heat equation with training points resampling:

\frac{\partial u}{\partial t}=\alpha \frac{\partial^2u}{\partial x^2}, \qquad x \in [-1, 1], \quad t \in [0, 1]

where \alpha=0.4 is the thermal diffusivity constant.

With Dirichlet boundary conditions:

u(0,t) = u(1,t)=0,

and periodic(sinusoidal) inital condition:

u(x,0) = \sin (\frac{n\pi x}{L}),\qquad 0<x<L, \quad n = 1,2,.....

where L=1 is the length of the bar, n=1 is the frequency of the sinusoidal initial conditions.

The exact solution is u(x,t) = e^{\frac{-n ^2\pi ^2 \alpha t}{L^2}}\sin (\frac{n\pi x}{L}).

Implementation

This description goes through the implementation of a solver for the above described Heat equation step-by-step.

First, the DeepXDE are imported:

import deepxde as dde

We begin by defining the parameters of the equation:

a = 0.4
L = 1
n = 1

Next, we define a computational geometry and time domain. We can use a built-in class Interval and TimeDomain and we combine both the domains using GeometryXTime as follows

geom = dde.geometry.Interval(0, L)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

Next, we express the PDE residual of the Heat equation:

def pde(x, y):
    dy_t = dde.grad.jacobian(y, x, i=0, j=1)
    dy_xx = dde.grad.hessian(y, x, i=0, j=0)
    return dy_t - a * dy_xx

The first argument to pde is 2-dimensional vector where the first component(x[:,0]) is x-coordinate and the second componenet (x[:,1]) is the t-coordinate. The second argument is the network output, i.e., the solution u(x,t), but here we use y as the name of the variable.

Next, we consider the boundary/initial condition. on_boundary is chosen here to use the whole boundary of the computational domain in considered as the boundary condition. We include the geomtime space, time geometry created above and on_boundary as the BCs in the DirichletBC function of DeepXDE. We also define IC which is the inital condition for the burgers equation and we use the computational domain, initial function, and on_initial to specify the IC.

bc = dde.icbc.DirichletBC(geomtime, lambda x: 0, lambda _, on_boundary: on_boundary)
ic = dde.icbc.IC(
    geomtime,
    lambda x: np.sin(n * np.pi * x[:, 0:1] / L),
    lambda _, on_initial: on_initial,
)

Now, we have specified the geometry, PDE residual, and boundary/initial condition. We then define the TimePDE problem as

data = dde.data.TimePDE(
    geomtime,
    pde,
    [bc, ic],
    num_domain=2540,
    num_boundary=80,
    num_initial=160,
    num_test=2540,
)

The number 2540 is the number of training residual points sampled inside the domain, and the number 80 is the number of training points sampled on the boundary. We also include 160 initial residual points for the initial conditions.

Next, we choose the network. Here, we use a fully connected neural network of depth 4 (i.e., 3 hidden layers) and width 20:

net = dde.nn.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")

Now, we have the PDE problem and the network. We build a Model and choose the optimizer and learning rate:

model = dde.Model(data, net)
model.compile("adam", lr=1e-3)

The following code is to apply mini-batch gradient descent sampling method. The period is the period of resamping. Here, the training points in the domain will be resampled every 10 iterations.

pde_resampler = dde.callbacks.PDEPointResampler(period=10)

We then train the model for 20000 iterations:

losshistory, train_state = model.train(iterations=200000, callbacks=[pde_resampler])

After we train the network using Adam, we continue to train the network using L-BFGS to achieve a smaller loss:

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

Complete code

.. literalinclude:: ../../../examples/pinn_forward/heat_resample.py
  :language: python