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

examples/pinn_forward/diffusion_1d.py

@@ -1,4 +1,14 @@
-"""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).
+
+Physical context: This represents a 1D rod with a decaying heat source, where the ends are kept at zero temperature (Dirichlet Boundary Conditions).
+"""
+
 import deepxde as dde
 import numpy as np
 # Backend tensorflow.compat.v1 or tensorflow
@@ -18,27 +28,34 @@ def pde(x, y):
     # 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 = tf.exp(-x[:, 1:]) * (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
+
+   # Backend tensorflow.compat.v1 or tensorflow
     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