🛠️ added pdf

PINNs_1DHeatEquationExample.ipynb

@@ -21,7 +21,7 @@
     "\n",
     "This notebook is based on two papers: *[Physics-Informed Neural Networks:  A Deep LearningFramework for Solving Forward and Inverse ProblemsInvolving Nonlinear Partial Differential Equations](https://www.sciencedirect.com/science/article/pii/S0021999118307125)* and *[Hidden Physics Models:  Machine Learning of NonlinearPartial Differential Equations](https://www.sciencedirect.com/science/article/pii/S0021999117309014)* with the help of  Fergus Shone and Michael Macraild.\n",
     "\n",
-    "These tutorials will go through solving Partial Differential Equations using Physics Informed Neuaral Networks focusing on the Burgers Equation and a more complex example using the Navier Stokes Equation\n",
+    "These tutorials will go through solving Partial Differential Equations using Physics Informed Neuaral Networks focusing on the 1D Heat Equation and a more complex example using the Navier Stokes Equation\n",
     "\n",
     "**This introduction section is replicated in all PINN tutorial notebooks (please skip if you've already been through)** \n",
     "\n",
@@ -93,15 +93,15 @@
     "Neural networks work by using lots of data to calculate weights and biases from data alone to minimise the loss function enabling them to act as universal fuction approximators. However these loose their robustness when data is limited. However by using know physical laws or empirical validated relationships the solutions from neural networks can be sufficiently constrianed by disregardins no realistic solutions.\n",
     "    \n",
     "A Physics Informed Nueral Network considers a parameterized and nonlinear partial differential equation in the genral form;\n",
-    "$$\n",
+    "\n",
     "\\begin{align}\n",
-    "     u_t + \\mathcal{N}[u; \\lambda] &= 0, && x \\in \\Omega, t \\in [0,T],\\\\\n",
+    "u_t + \\mathcal{N}[u; \\lambda]  = 0,  x \\in \\Omega, t \\in [0,T],\\\\\n",
     "\\end{align}\n",
-    "$$\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 perscribed data). This set up an encapuslate 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 Burgers equation and Navier stokes equations\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 perscribed data). This set up an encapuslate 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 1D headt equation and Navier stokes equations\n",
     "\n",
     "\n",
     "</div>    "