add example and update faq to include mixed precision

Author: g-w1

docs/user/faq.rst

@@ -10,6 +10,8 @@ General usage
   | **A**: `#5`_
 - | **Q**: By default, DeepXDE uses ``float32``. How can I use ``float64``?
   | **A**: `#28`_
+- | **Q**: How can I use mixed precision training?
+  | **A**: Use ``dde.config.set_default_float("mixed")`` with the ``tensorflow`` or ``pytorch`` backends. See `this paper <https://arxiv.org/abs/2401.16645>`_ for more information.
 - | **Q**: I want to set the global random seeds.
   | **A**: `#353`_
 - | **Q**: GPU.

examples/pinn_forward/Burgers_mixed.py

@@ -0,0 +1,55 @@
+"""Backend supported: tensorflow, pytorch
+The exact same as Burgers.py, but using mixed precision instead of float32.
+This preserves accuracy while speeding up training (especially with larger training runs).
+"""
+
+import deepxde as dde
+import numpy as np
+
+dde.config.set_default_float("mixed")
+
+
+def gen_testdata():
+    data = np.load("../dataset/Burgers.npz")
+    t, x, exact = data["t"], data["x"], data["usol"].T
+    xx, tt = np.meshgrid(x, t)
+    X = np.vstack((np.ravel(xx), np.ravel(tt))).T
+    y = exact.flatten()[:, None]
+    return X, y
+
+
+def pde(x, y):
+    dy_x = dde.grad.jacobian(y, x, i=0, j=0)
+    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 + y * dy_x - 0.01 / np.pi * dy_xx
+
+
+geom = dde.geometry.Interval(-1, 1)
+timedomain = dde.geometry.TimeDomain(0, 0.99)
+geomtime = dde.geometry.GeometryXTime(geom, timedomain)
+
+bc = dde.icbc.DirichletBC(geomtime, lambda x: 0, lambda _, on_boundary: on_boundary)
+ic = dde.icbc.IC(
+    geomtime, lambda x: -np.sin(np.pi * x[:, 0:1]), lambda _, on_initial: on_initial
+)
+
+data = dde.data.TimePDE(
+    geomtime, pde, [bc, ic], num_domain=2540, num_boundary=80, num_initial=160
+)
+net = dde.nn.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
+model = dde.Model(data, net)
+
+model.compile("adam", lr=1e-3)
+losshistory, train_state = model.train(iterations=15000)
+# We have to disable L-BFGS since it does not support mixed precision
+# model.compile("L-BFGS")
+# losshistory, train_state = model.train()
+dde.saveplot(losshistory, train_state, issave=True, isplot=True)
+
+X, y_true = gen_testdata()
+y_pred = model.predict(X)
+f = model.predict(X, operator=pde)
+print("Mean residual:", np.mean(np.absolute(f)))
+print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
+np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))