[doc] Update docs (#4)

* Update

* Update

* Find bug in Allen_Cahn.ipynb

* Update

* Update

* Update heat.py

* Update

* Update Burgers_RAR.py

* Fix heat_resample.py

* Fix bugs in `heat_resample.py`

* Fix `Laplace_disk.py`

* Add `Burgers_RAR.ipynb`

* Add `heat_resample.ipynb`

* Add `Laplace_disk.ipynb`

.gitignore

@@ -25,3 +25,7 @@ pinnx/_version.py
 .idea/
 
 related-projects.md
+
+# .dat files in the docs folder and it's subfolders
+docs/**/*.dat
+docs/**/*.npz

docs/examples-function/dataset.py

@@ -12,14 +12,16 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 # ==============================================================================
+import os
 
 import brainstate as bst
 import numpy as np
 
 import pinnx
 
-train_data = np.loadtxt("../dataset/dataset.train")
-test_data = np.loadtxt("../dataset/dataset.test")
+PATH = os.path.dirname(os.path.abspath(__file__))
+train_data = np.loadtxt(os.path.join(PATH, '..', 'dataset', 'dataset.train'))
+test_data = np.loadtxt(os.path.join(PATH, '..', 'dataset', 'dataset.test'))
 
 net = pinnx.nn.Model(
     pinnx.nn.DictToArray(x=None),

docs/examples-pinn-forward/Burgers_RAR.py

@@ -48,7 +48,7 @@ def pde(x, y):
 
 X = geomtime.random_points(100000)
 err = 1
-while u.get_magnitude(err) > 0.005:
+while u.get_magnitude(err) > 0.012:
     f = trainer.predict(X, operator=pde)
     err_eq = u.math.absolute(f)
     err = u.math.mean(err_eq)

docs/examples-pinn-forward/Euler_beam.ipynb

@@ -0,0 +1,44 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Euler beam\n",
+    "## Problem setup\n",
+    "\n",
+    "We will solve a Euler beam problem:\n",
+    "$$\n",
+    "\\frac{\\partial^{4} u}{\\partial x^4} + 1 = 0, \\qquad x \\in [0, 1],\n",
+    "$$\n",
+    "with two boundary conditions on the right boundary,\n",
+    "\n",
+    "$$\n",
+    "u''(1)=0,   u'''(1)=0\n",
+    "$$\n",
+    "\n",
+    "and one Dirichlet boundary condition on the left boundary,\n",
+    "\n",
+    "$$\n",
+    "u(0)=0\n",
+    "$$\n",
+    "\n",
+    "along with one Neumann boundary condition on the left boundary,\n",
+    "\n",
+    "$$\n",
+    "u'(0)=0\n",
+    "$$\n",
+    "\n",
+    "The exact solution is $u(x) = -\\frac{1}{24}x^4+\\frac{1}{6}x^3-\\frac{1}{4}x^2.$\n",
+    "\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

docs/examples-pinn-forward/Helmholtz_Dirichlet_2d.ipynb

@@ -0,0 +1,44 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Helmholtz equation over a 2D square domain\n",
+    "\n",
+    "## Problem setup\n",
+    "For a wavenumber $k_0 = 2\\pi n$ with $n = 2$, we will solve a Helmholtz equation:\n",
+    "\n",
+    "$$\n",
+    "- u_{xx}-u_{yy} - k_0^2 u = f, \\qquad  \\Omega = [0,1]^2\n",
+    "$$\n",
+    "\n",
+    "with the Dirichlet boundary conditions\n",
+    "\n",
+    "$$\n",
+    "u(x,y)=0, \\qquad (x,y)\\in \\partial \\Omega\n",
+    "$$\n",
+    "\n",
+    "and a source term $f(x,y) = k_0^2 \\sin(k_0 x)\\sin(k_0 y)$.\n",
+    "\n",
+    "Remark that the exact solution reads:\n",
+    "$$\n",
+    "u(x,y)= \\sin(k_0 x)\\sin(k_0 y)\n",
+    "$$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "pinnx",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "name": "python",
+   "version": "3.10.15"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

docs/examples-pinn-forward/Klein_Gordon.ipynb

@@ -0,0 +1,52 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Klein-Gordon equation\n",
+    "\n",
+    "## Problem setup\n",
+    "\n",
+    "We will solve a Klein-Gordon equation:\n",
+    "\n",
+    "$$\n",
+    "\\frac{\\partial^2y}{\\partial t^2} + \\alpha \\frac{\\partial^2y}{\\partial x^2} + \\beta y + \\gamma y^k = -x\\cos(t) + x^2\\cos^2(t), \\qquad x \\in [-1, 1], \\quad t \\in [0, 10]\n",
+    "$$\n",
+    "\n",
+    "with initial conditions\n",
+    "\n",
+    "$$\n",
+    "y(x, 0) = x, \\quad \\frac{\\partial y}{\\partial t}(x, 0) = 0\n",
+    "$$\n",
+    "\n",
+    "and Dirichlet boundary conditions\n",
+    "\n",
+    "$$\n",
+    "y(-1, t) = -\\cos(t), \\quad y(1, t) = \\cos(t)\n",
+    "$$\n",
+    "\n",
+    "We also specify the following parameters for the equation:\n",
+    "\n",
+    "$$\n",
+    "\\alpha = -1, \\beta = 0, \\gamma = 1, k = 2.\n",
+    "$$\n",
+    "\n",
+    "The reference solution is $y(x, t) = x\\cos(t)$."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "pinnx",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "name": "python",
+   "version": "3.10.15"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

docs/examples-pinn-forward/Kovasznay_flow.ipynb

@@ -0,0 +1,44 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Kovasznay flow\n",
+    "\n",
+    "## Problem setup\n",
+    "\n",
+    "We will solve the Kovasznay flow equation on $\\Omega = [0, 1]^2$:\n",
+    "\n",
+    "$$\n",
+    "u\\frac{\\partial u}{\\partial x} + v\\frac{\\partial u}{\\partial y}= -\\frac{\\partial p}{\\partial x} + \\frac{1}{Re}(\\frac{\\partial^2u}{\\partial x^2} + \\frac{\\partial^2u}{\\partial y^2}),\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "u\\frac{\\partial v}{\\partial x} + v\\frac{\\partial v}{\\partial y}= -\\frac{\\partial p}{\\partial y} + \\frac{1}{Re}(\\frac{\\partial^2v}{\\partial x^2} + \\frac{\\partial^2v}{\\partial y^2}),\n",
+    "$$\n",
+    "\n",
+    "with the Dirichlet boundary conditions\n",
+    "\n",
+    "$$\n",
+    "u(x,y)=0, \\qquad (x,y)\\in \\partial \\Omega\n",
+    "$$\n",
+    "\n",
+    "The reference solution is $u = 1 - e^{\\lambda x} \\cos(2\\pi y), v = \\frac{\\lambda}{2\\pi}e^{\\lambda x} \\sin(2\\pi x)$, $p =\\frac{1}{2}(1 - e^{2\\lambda x})$, where $\\lambda = \\frac{1}{2\\nu}-\\sqrt{\\frac{1}{4\\nu^2}+4\\pi^2}$."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "pinnx",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "name": "python",
+   "version": "3.10.15"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}