🛠️ added pdf

PINNs_1DHeatEquation_nonML.ipynb

@@ -93,15 +93,15 @@
     "A Physics Informed Neural Network considers a parameterized and nonlinear partial differential equation in the general form;\n",
     "\n",
     "\n",
-    "\\begin{equation}\n",
+    "\n",
     "    \n",
     "\\begin{align}\n",
     "     u_t + \\mathcal{N}[u; \\lambda] &= 0, && x \\in \\Omega, t \\in [0,T],\\\\\n",
     "\\end{align}\n",
     "    \n",
-    "\\end{equation}\n",
     "\n",
-    "where $\\mathcal{u(t,x)}$ denores the hidden solution, $\\mathcal{N}$ is a nonlinear differential operator acting on $u$, $\\mathcal{\\lambda}$ and $\\Omega$ is a \\subset of \\mathbb{R}^D$ (the prescribed data). This set up an encapsulation of a wide range of problems such as diffusion processes, conservation laws,  advection-diffusion-reaction  systems,  and  kinetic  equations and conservation laws.\n",
+    "\n",
+    "where $\\mathcal{u(t,x)}$ denores the hidden solution, $\\mathcal{N}$ is a nonlinear differential operator acting on $u$, $\\mathcal{\\lambda}$ and $\\Omega$ is a subset of $\\mathbb{R}^D$ (the prescribed data). This set up an encapsulation of a wide range of problems such as diffusion processes, conservation laws,  advection-diffusion-reaction  systems,  and  kinetic  equations and conservation laws.\n",
     "\n",
     "Here we will go though this for the 1 Heat equation and Navier stokes equations\n",
     "\n",
@@ -326,15 +326,15 @@
     "\n",
     "After simple algebraic manipulations and exploiting the initial condition $u(x,0) = m(x)$, we then obtain\n",
     "\n",
+    "\n",
+    "\n",
     "\\begin{equation}\n",
-    "\\left\\{\n",
-    "\\begin{array}\n",
     "\\mathbf{u}^{(0)} = \\mathbf{m} \\\\\n",
     "\\mathbf{u}^{(i+1)} = \\left( I + \\Delta t\\, K\\right)^{-1} \\mathbf{u}^{(i)},\n",
-    "\\end{array}\n",
-    "\\right.\n",
     "\\end{equation}\n",
     "\n",
+    "\n",
+    "\n",
     "or equivalently\n",
     "\n",
     "\\begin{equation} \\mathbf{u}^{(i)} = \\left( I + \\Delta t\\, K\\right)^{-i} \\mathbf{m}.\\end{equation}\n",
@@ -535,7 +535,9 @@
     "    \n",
     "### Why does the naive solution fail?\n",
     "\n",
-    "Let \\begin{equation}v_n = \\sqrt{\\frac{2}{L}} \\sin\\left( n \\, \\frac{\\pi}{L} x \\right)$ with $n=1,2,3, \\ldots \\end{equation}, then we have that\n",
+    "Let \\begin{equation}v_n = \\sqrt{\\frac{2}{L}} \\sin\\left( n \\, \\frac{\\pi}{L} x \\right)\\end{equation} \n",
+    "\n",
+    "with n=1,2,3, then we have that\n",
     "\n",
     "\\begin{equation} \\mathcal{F} v_n = \\lambda_n v_n, \\quad \\text{where the eigenvalues } \\lambda_n = e^{-kT\\left(\\frac{\\pi}{L} n \\right)^2}. \\end{equation}\n",
     "\n",