Add physical context to examples, and use backend-agnostic functions

examples/pinn_forward/diffusion_1d.py

@@ -1,4 +1,10 @@
-"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle"""
+"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle
+1D Diffusion Equation with a Time-Dependent Source Term.
+
+This example solves the heat equation:
+∂y/∂t - ∂²y/∂x² = f(x, t)
+where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx).
+"""
 import deepxde as dde
 import numpy as np
 # Backend tensorflow.compat.v1 or tensorflow
@@ -15,37 +21,17 @@ def pde(x, y):
     # Most backends
     dy_t = dde.grad.jacobian(y, x, j=1)
     dy_xx = dde.grad.hessian(y, x, j=0)
-    # Backend jax
-    # dy_t, _ = dde.grad.jacobian(y, x, j=1)
-    # dy_xx, _ = dde.grad.hessian(y, x, j=0)
-    # Backend tensorflow.compat.v1 or tensorflow
+
+    # Physics Note: The following term is the forced heat source f(x, t)
+    # required to satisfy the analytical solution y = e^(-t)sin(πx).
+    source_term_val = dde.backend.exp(-x[:, 1:]) * (
+        dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1])
+    )
     return (
         dy_t
         - dy_xx
-        + tf.exp(-x[:, 1:])
-        * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
-    )
-    # Backend pytorch
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + torch.exp(-x[:, 1:])
-    #     * (torch.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * torch.sin(np.pi * x[:, 0:1]))
-    # )
-    # Backend jax
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + jnp.exp(-x[:, 1:])
-    #     * (jnp.sin(np.pi * x[..., 0:1]) - np.pi ** 2 * jnp.sin(np.pi * x[..., 0:1]))
-    # )
-    # Backend paddle
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + paddle.exp(-x[:, 1:])
-    #     * (paddle.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * paddle.sin(np.pi * x[:, 0:1]))
-    # )
+        + source_term_val
+    ) 
 
 
 def func(x):

examples/pinn_forward/diffusion_1d_exactBC.py

@@ -1,4 +1,10 @@
-"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle"""
+"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle
+1D Diffusion Equation with a Time-Dependent Source Term.
+
+This example solves the heat equation:
+∂y/∂t - ∂²y/∂x² = f(x, t)
+where the source term f(x, t) is chosen such that the analytical solution is y = e^(-t) * sin(πx).
+"""
 import deepxde as dde
 import numpy as np
 # Backend tensorflow.compat.v1 or tensorflow
@@ -15,37 +21,17 @@ def pde(x, y):
     # Most backends
     dy_t = dde.grad.jacobian(y, x, i=0, j=1)
     dy_xx = dde.grad.hessian(y, x, i=0, j=0)
-    # Backend jax
-    # dy_t, _ = dde.grad.jacobian(y, x, i=0, j=1)
-    # dy_xx, _ = dde.grad.hessian(y, x, i=0, j=0)
-    # Backend tensorflow.compat.v1 or tensorflow
+
+    # Physics Note: The following term is the forced heat source f(x, t)
+    # required to satisfy the analytical solution y = e^(-t)sin(πx).
+    source_term_val = dde.backend.exp(-x[:, 1:]) * (
+        dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1])
+    )
     return (
         dy_t
         - dy_xx
-        + tf.exp(-x[:, 1:])
-        * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
+        + source_term_val
     )
-    # Backend pytorch
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + torch.exp(-x[:, 1:])
-    #     * (torch.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * torch.sin(np.pi * x[:, 0:1]))
-    # )
-    # Backend jax
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + jnp.exp(-x[:, 1:])
-    #     * (jnp.sin(np.pi * x[..., 0:1]) - np.pi ** 2 * jnp.sin(np.pi * x[..., 0:1]))
-    # )
-    # Backend paddle
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + paddle.exp(-x[:, 1:])
-    #     * (paddle.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * paddle.sin(np.pi * x[:, 0:1]))
-    # )
 
 
 def func(x):
@@ -63,14 +49,7 @@ def func(x):
 initializer = "Glorot uniform"
 net = dde.nn.FNN(layer_size, activation, initializer)
 net.apply_output_transform(
-    # Backend tensorflow.compat.v1 or tensorflow
-    lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + tf.sin(np.pi * x[:, 0:1])
-    # Backend pytorch
-    # lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + torch.sin(np.pi * x[:, 0:1])
-    # Backend jax
-    # lambda x, y: x[..., 1:2] * (1 - x[..., 0:1] ** 2) * y + jnp.sin(np.pi * x[..., 0:1])
-    # Backend paddle
-    # lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + paddle.sin(np.pi * x[:, 0:1])
+    lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + dde.backend.sin(np.pi * x[:, 0:1])
 )
 
 model = dde.Model(data, net)

examples/pinn_forward/diffusion_1d_resample.py

@@ -1,4 +1,10 @@
-"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle"""
+"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, jax, paddle
+1D Diffusion Equation with Adaptive Point Resampling.
+
+This example solves the heat equation using the PDEPointResampler callback:
+∂y/∂t - ∂²y/∂x² = f(x, t)
+Analytical solution: y = e^(-t) * sin(πx).
+"""
 import deepxde as dde
 import numpy as np
 # Backend tensorflow.compat.v1 or tensorflow
@@ -15,39 +21,19 @@ def pde(x, y):
     # Most backends
     dy_t = dde.grad.jacobian(y, x, i=0, j=1)
     dy_xx = dde.grad.hessian(y, x, i=0, j=0)
-    # Backend jax
-    # dy_t, _ = dde.grad.jacobian(y, x, i=0, j=1)
-    # dy_xx, _ = dde.grad.hessian(y, x, i=0, j=0)
-    # Backend tensorflow.compat.v1 or tensorflow
+
+    # Physics Note: The following term is the forced heat source f(x, t)
+    # required to satisfy the analytical solution y = e^(-t)sin(πx).
+    source_term_val = dde.backend.exp(-x[:, 1:]) * (
+        dde.backend.sin(np.pi * x[:, 0:1]) - np.pi**2 * dde.backend.sin(np.pi * x[:, 0:1])
+    )
     return (
         dy_t
         - dy_xx
-        + tf.exp(-x[:, 1:])
-        * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
+        + source_term_val
     )
-    # Backend pytorch
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + torch.exp(-x[:, 1:])
-    #     * (torch.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * torch.sin(np.pi * x[:, 0:1]))
-    # )
-    # Backend jax
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + jnp.exp(-x[:, 1:])
-    #     * (jnp.sin(np.pi * x[..., 0:1]) - np.pi ** 2 * jnp.sin(np.pi * x[..., 0:1]))
-    # )
-    # Backend paddle
-    # return (
-    #     dy_t
-    #     - dy_xx
-    #     + paddle.exp(-x[:, 1:])
-    #     * (paddle.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * paddle.sin(np.pi * x[:, 0:1]))
-    # )
-
-
+
+
 def func(x):
     return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])
 
@@ -77,7 +63,11 @@ def func(x):
 
 model = dde.Model(data, net)
 
+# Adaptive Resampling Callback
+# Period=100 means every 100 iterations, the model redistributes 
+# the domain points to where the PDE residual is highest.
 resampler = dde.callbacks.PDEPointResampler(period=100)
+
 model.compile("adam", lr=0.001, metrics=["l2 relative error"])
 losshistory, train_state = model.train(iterations=2000, callbacks=[resampler])