Add forward wave equation example

examples/wave_equation/readme.md

@@ -0,0 +1,16 @@
+# Forward Wave Equation
+
+Solve the 1D wave equation using IDRLnet:
+
+$$u_{tt} = c^2 u_{xx}$$
+
+on $[0, 1] \times [0, 1]$ with fixed-end boundary conditions, sinusoidal initial
+displacement, and zero initial velocity.
+
+**Exact solution:** $u(x, t) = \sin(\pi x) \cos(\pi c t)$
+
+## Usage
+
+```bash
+python wave_equation.py
+```

examples/wave_equation/wave_equation.py

@@ -0,0 +1,133 @@
+"""Forward 1D wave equation solved with IDRLnet.
+
+PDE:  u_tt = c^2 * u_xx  on [0, 1] x [0, 1]
+BCs:  u(0, t) = u(1, t) = 0  (fixed ends)
+ICs:  u(x, 0) = sin(pi * x),  u_t(x, 0) = 0
+Exact solution: u(x, t) = sin(pi * x) * cos(pi * c * t)
+"""
+
+from sympy import Symbol, sin, cos, pi
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.tri as tri
+import idrlnet.shortcut as sc
+
+c = 1.0  # wave speed
+
+x = Symbol("x")
+t_symbol = Symbol("t")
+time_range = {t_symbol: (0, 1)}
+geo = sc.Line1D(0.0, 1.0)
+
+
+@sc.datanode(name="wave_equation")
+def interior_domain():
+    points = geo.sample_interior(
+        10000, bounds={x: (0.0, 1.0)}, param_ranges=time_range
+    )
+    constraints = {"wave_equation": 0}
+    return points, constraints
+
+
+@sc.datanode(name="t_init")
+def init_domain():
+    """Initial displacement: u(x, 0) = sin(pi * x)."""
+    points = geo.sample_interior(200, param_ranges={t_symbol: 0.0})
+    constraints = sc.Variables({"u": sin(pi * x)})
+    return points, constraints
+
+
+@sc.datanode(name="t_init_vel")
+def init_velocity_domain():
+    """Initial velocity: u_t(x, 0) = 0."""
+    points = geo.sample_interior(200, param_ranges={t_symbol: 0.0})
+    constraints = sc.Variables({"u__t": 0.0})
+    return points, constraints
+
+
+@sc.datanode(name="x_boundary")
+def boundary_domain():
+    """Boundary conditions: u(0, t) = u(1, t) = 0."""
+    points = geo.sample_boundary(200, param_ranges=time_range)
+    constraints = sc.Variables({"u": 0.0})
+    return points, constraints
+
+
+net = sc.get_net_node(
+    inputs=("x", "t"),
+    outputs=("u",),
+    name="net1",
+    arch=sc.Arch.mlp,
+)
+pde = sc.WaveNode(c=c, dim=1, time=True, u="u")
+s = sc.Solver(
+    sample_domains=(
+        interior_domain(),
+        init_domain(),
+        init_velocity_domain(),
+        boundary_domain(),
+    ),
+    netnodes=[net],
+    pdes=[pde],
+    max_iter=5000,
+)
+s.solve()
+
+# Inference and plotting
+coord = s.infer_step(
+    {
+        "wave_equation": ["x", "t", "u"],
+        "t_init": ["x", "t"],
+        "x_boundary": ["x", "t"],
+    }
+)
+num_x = coord["wave_equation"]["x"].cpu().detach().numpy().ravel()
+num_t = coord["wave_equation"]["t"].cpu().detach().numpy().ravel()
+num_u = coord["wave_equation"]["u"].cpu().detach().numpy().ravel()
+
+init_x = coord["t_init"]["x"].cpu().detach().numpy().ravel()
+init_t = coord["t_init"]["t"].cpu().detach().numpy().ravel()
+boundary_x = coord["x_boundary"]["x"].cpu().detach().numpy().ravel()
+boundary_t = coord["x_boundary"]["t"].cpu().detach().numpy().ravel()
+
+# Exact solution for comparison
+u_exact = np.sin(np.pi * num_x) * np.cos(np.pi * c * num_t)
+
+triang_total = tri.Triangulation(num_t, num_x)
+
+fig, axes = plt.subplots(1, 3, figsize=(18, 5))
+
+# Predicted solution
+ax1 = axes[0]
+tcf = ax1.tricontourf(triang_total, num_u, 100, cmap="jet")
+plt.colorbar(tcf, ax=ax1)
+ax1.set_xlabel("$t$")
+ax1.set_ylabel("$x$")
+ax1.set_title("Predicted $u(x,t)$")
+ax1.scatter(init_t, init_x, c="black", marker="x", s=8)
+ax1.scatter(boundary_t, boundary_x, c="black", marker="x", s=8)
+
+# Exact solution
+ax2 = axes[1]
+tcf2 = ax2.tricontourf(triang_total, u_exact, 100, cmap="jet")
+plt.colorbar(tcf2, ax=ax2)
+ax2.set_xlabel("$t$")
+ax2.set_ylabel("$x$")
+ax2.set_title("Exact $u(x,t)$")
+
+# Absolute error
+ax3 = axes[2]
+tcf3 = ax3.tricontourf(triang_total, np.abs(num_u - u_exact), 100, cmap="jet")
+plt.colorbar(tcf3, ax=ax3)
+ax3.set_xlabel("$t$")
+ax3.set_ylabel("$x$")
+ax3.set_title("Absolute error")
+
+plt.tight_layout()
+plt.savefig("wave_equation.png", dpi=500, bbox_inches="tight", pad_inches=0.02)
+plt.show()
+plt.close()
+
+# Print L2 error
+l2_error = np.sqrt(np.mean((num_u - u_exact) ** 2))
+print(f"L2 error: {l2_error:.6f}")