One-dimensional Diffusion Equation#
Problem setup#
We will solve a diffusion equation:
with the initial condition
and the Dirichlet boundary condition
The reference solution is \(y = e^{-t} \sin(\pi x)\).
Dimensional Analysis#
Below is the dimensional analysis of the given diffusion equation with an unknown parameter \(C\):
Step 1: Assign Dimensions to Variables#
Spatial Coordinate \(x\):
Spatial coordinate has the dimension of length:
\[ [x] = L. \]
Time \(t\):
Time has the dimension:
\[ [t] = T. \]
Function \(y(x, t)\):
\(y\) is the solution of the diffusion equation and depends on the context. For this case, \(y\) has no explicit physical quantity associated with it, but we assume it to be dimensionless since the reference solution is given as \( y = e^{-t} \sin(\pi x) \), where both \(e^{-t}\) and \(\sin(\pi x)\) are dimensionless.
\[ [y] = 1 \quad \text{(dimensionless)}. \]
Parameter \(C\):
The term \(C \frac{\partial^2 y}{\partial x^2}\) must have the same dimension as \(\frac{\partial y}{\partial t}\) for consistency.
First, consider the time derivative:
\[ \left[\frac{\partial y}{\partial t}\right] = \frac{[y]}{[t]} = \frac{1}{T}. \]Next, consider the second spatial derivative:
\[ \left[\frac{\partial^2 y}{\partial x^2}\right] = \frac{[y]}{[x]^2} = \frac{1}{L^2}. \]Multiplying by \(C\), the dimensions of \(C\) must satisfy:
\[ [C] \cdot \frac{1}{L^2} = \frac{1}{T} \implies [C] = \frac{L^2}{T}. \]
Source Term: \(e^{-t} \left(\sin(\pi x) - \pi^2 \sin(\pi x)\right)\)
The exponential term \(e^{-t}\) and the sine functions are dimensionless. Therefore, the source term is dimensionally consistent with:
\[ \text{Source Term} = \frac{1}{T}. \]
Step 2: Initial and Boundary Conditions#
Initial Condition: \(y(x, 0) = \sin(\pi x)\).
\(\sin(\pi x)\) is dimensionless, consistent with \( [y] = 1 \).
Boundary Condition: \(y(-1, t) = y(1, t) = 0\).
The boundary values are dimensionless.
Step 3: Summary of Dimensions#
Variable/Parameter |
Physical Meaning |
Dimensions |
|---|---|---|
\(x\) |
Spatial coordinate |
\(L\) |
\(t\) |
Time |
\(T\) |
\(y\) |
Solution (dimensionless) |
\(1\) |
\(C\) |
Diffusion coefficient |
\(L^2 / T\) |
Source term |
Forcing function |
\(1 / T\) |
In conclusion,
The unknown parameter \(C\) has dimensions of \(L^2 / T\), which is consistent with the physical meaning of a diffusion coefficient.
The function \(y\) and the boundary/initial conditions are dimensionless, ensuring the consistency of the problem setup.
Implementation#
Import the required libraries:
import braintools
import brainunit as u
import pinnx
Define the physical units for the problem:
unit_of_x = u.meter
unit_of_t = u.second
unit_of_f = 1 / u.second
c = 1. * u.meter ** 2 / u.second
Define the geometry and time domain:
geom = pinnx.geometry.Interval(-1, 1)
timedomain = pinnx.geometry.TimeDomain(0, 1)
geomtime = pinnx.geometry.GeometryXTime(geom, timedomain)
geomtime = geomtime.to_dict_point(x=unit_of_x, t=unit_of_t)
Define the initial condition and boundary condition functions:
def func(x):
y = u.math.sin(u.math.pi * x['x'] / unit_of_x) * u.math.exp(-x['t'] / unit_of_t)
return {'y': y}
bc = pinnx.icbc.DirichletBC(func)
ic = pinnx.icbc.IC(func)
Define the neural network model:
net = pinnx.nn.Model(
pinnx.nn.DictToArray(x=unit_of_x, t=unit_of_t),
pinnx.nn.FNN([2] + [32] * 3 + [1], "tanh"),
pinnx.nn.ArrayToDict(y=None),
)
Define the PDE function:
def pde(x, y):
jacobian = net.jacobian(x, x='t')
hessian = net.hessian(x, xi='x', xj='x')
dy_t = jacobian["y"]["t"]
dy_xx = hessian["y"]["x"]["x"]
source = (
u.math.exp(-x['t'] / unit_of_t) * (
u.math.sin(u.math.pi * x['x'] / unit_of_x) -
u.math.pi ** 2 * u.math.sin(u.math.pi * x['x'] / unit_of_x)
)
)
return dy_t - c * dy_xx + source * unit_of_f
Define the problem and train the model:
problem = pinnx.problem.TimePDE(
geomtime,
pde,
[bc, ic],
net,
num_domain=40,
num_boundary=20,
num_initial=10,
solution=func,
num_test=10000,
)
Warning: 10000 points required, but 10082 points sampled.
Train the model:
trainer = pinnx.Trainer(problem)
trainer.compile(braintools.optim.Adam(0.001), metrics=["l2 relative error"]).train(iterations=10000)
trainer.saveplot(issave=True, isplot=True)
Compiling trainer...
'compile' took 0.066982 s
Training trainer...
Step Train loss Test loss Test metric
0 [39.28094 * becquerel2, [42.6258 * becquerel2, [{'y': Array(2.4083016, dtype=float32)}]
{'ibc0': {'y': Array(0.82120794, dtype=float32)}}, {'ibc0': {'y': Array(0.82120794, dtype=float32)}},
{'ibc1': {'y': Array(1.7334808, dtype=float32)}}] {'ibc1': {'y': Array(1.7334808, dtype=float32)}}]
1000 [0.00716378 * becquerel2, [0.03221128 * becquerel2, [{'y': Array(0.08175115, dtype=float32)}]
{'ibc0': {'y': Array(0.00580588, dtype=float32)}}, {'ibc0': {'y': Array(0.00580588, dtype=float32)}},
{'ibc1': {'y': Array(0.00591157, dtype=float32)}}] {'ibc1': {'y': Array(0.00591157, dtype=float32)}}]
2000 [0.00374766 * becquerel2, [0.01406009 * becquerel2, [{'y': Array(0.0635821, dtype=float32)}]
{'ibc0': {'y': Array(0.0034853, dtype=float32)}}, {'ibc0': {'y': Array(0.0034853, dtype=float32)}},
{'ibc1': {'y': Array(0.00233263, dtype=float32)}}] {'ibc1': {'y': Array(0.00233263, dtype=float32)}}]
3000 [0.00207005 * becquerel2, [0.00866511 * becquerel2, [{'y': Array(0.04210193, dtype=float32)}]
{'ibc0': {'y': Array(0.00153627, dtype=float32)}}, {'ibc0': {'y': Array(0.00153627, dtype=float32)}},
{'ibc1': {'y': Array(0.00113975, dtype=float32)}}] {'ibc1': {'y': Array(0.00113975, dtype=float32)}}]
4000 [0.00109155 * becquerel2, [0.00996974 * becquerel2, [{'y': Array(0.02316353, dtype=float32)}]
{'ibc0': {'y': Array(0.00045823, dtype=float32)}}, {'ibc0': {'y': Array(0.00045823, dtype=float32)}},
{'ibc1': {'y': Array(0.00050199, dtype=float32)}}] {'ibc1': {'y': Array(0.00050199, dtype=float32)}}]
5000 [0.00051956 * becquerel2, [0.01253793 * becquerel2, [{'y': Array(0.01541764, dtype=float32)}]
{'ibc0': {'y': Array(0.00017889, dtype=float32)}}, {'ibc0': {'y': Array(0.00017889, dtype=float32)}},
{'ibc1': {'y': Array(0.00024536, dtype=float32)}}] {'ibc1': {'y': Array(0.00024536, dtype=float32)}}]
6000 [0.00026679 * becquerel2, [0.01434105 * becquerel2, [{'y': Array(0.01198213, dtype=float32)}]
{'ibc0': {'y': Array(0.00010253, dtype=float32)}}, {'ibc0': {'y': Array(0.00010253, dtype=float32)}},
{'ibc1': {'y': Array(0.00014029, dtype=float32)}}] {'ibc1': {'y': Array(0.00014029, dtype=float32)}}]
7000 [0.00015957 * becquerel2, [0.01430503 * becquerel2, [{'y': Array(0.00955142, dtype=float32)}]
{'ibc0': {'y': Array(6.4639426e-05, dtype=float32)}}, {'ibc0': {'y': Array(6.4639426e-05, dtype=float32)}},
{'ibc1': {'y': Array(8.0856196e-05, dtype=float32)}}] {'ibc1': {'y': Array(8.0856196e-05, dtype=float32)}}]
8000 [0.00011952 * becquerel2, [0.01355371 * becquerel2, [{'y': Array(0.00870061, dtype=float32)}]
{'ibc0': {'y': Array(5.1637177e-05, dtype=float32)}}, {'ibc0': {'y': Array(5.1637177e-05, dtype=float32)}},
{'ibc1': {'y': Array(4.7870093e-05, dtype=float32)}}] {'ibc1': {'y': Array(4.7870093e-05, dtype=float32)}}]
9000 [2.9594898e-05 * becquerel2, [0.01290757 * becquerel2, [{'y': Array(0.00810704, dtype=float32)}]
{'ibc0': {'y': Array(3.8509617e-05, dtype=float32)}}, {'ibc0': {'y': Array(3.8509617e-05, dtype=float32)}},
{'ibc1': {'y': Array(2.8898923e-05, dtype=float32)}}] {'ibc1': {'y': Array(2.8898923e-05, dtype=float32)}}]
10000 [1.6880738e-05 * becquerel2, [0.01241341 * becquerel2, [{'y': Array(0.00777998, dtype=float32)}]
{'ibc0': {'y': Array(3.39993e-05, dtype=float32)}}, {'ibc0': {'y': Array(3.39993e-05, dtype=float32)}},
{'ibc1': {'y': Array(2.0438354e-05, dtype=float32)}}] {'ibc1': {'y': Array(2.0438354e-05, dtype=float32)}}]
Best trainer at step 10000:
train loss: 7.13e-05
test loss: 1.25e-02
test metric: [{'y': Array(0.01, dtype=float32)}]
'train' took 8.116077 s
Saving loss history to D:\codes\projects\pinnx\docs\examples-pinn-forward\loss.dat ...
Saving checkpoint into D:\codes\projects\pinnx\docs\examples-pinn-forward\loss.dat
Saving training data to D:\codes\projects\pinnx\docs\examples-pinn-forward\train.dat ...
Saving checkpoint into D:\codes\projects\pinnx\docs\examples-pinn-forward\train.dat
Saving test data to D:\codes\projects\pinnx\docs\examples-pinn-forward\test.dat ...
Saving checkpoint into D:\codes\projects\pinnx\docs\examples-pinn-forward\test.dat
