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"Name: equinox\n",
"Version: 0.11.2\n",
"Summary: Elegant easy-to-use neural networks in JAX.\n",
"Home-page: \n",
"Author: \n",
"Author-email: Patrick Kidger \n",
"License: Apache License\n",
" Version 2.0, January 2004\n",
" http://www.apache.org/licenses/\n",
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"Location: /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages\n",
"Requires: jax, jaxtyping, typing-extensions\n",
"Required-by: diffrax, lineax, optimistix, orthojax\n",
"Requirement already satisfied: equinox in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (0.11.2)\n",
"Requirement already satisfied: jax>=0.4.13 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from equinox) (0.4.19)\n",
"Requirement already satisfied: jaxtyping>=0.2.20 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from equinox) (0.2.25)\n",
"Requirement already satisfied: typing-extensions>=4.5.0 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from equinox) (4.8.0)\n",
"Requirement already satisfied: ml-dtypes>=0.2.0 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (0.3.1)\n",
"Requirement already satisfied: numpy>=1.22 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (1.25.2)\n",
"Requirement already satisfied: opt-einsum in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (3.3.0)\n",
"Requirement already satisfied: scipy>=1.9 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (1.11.3)\n",
"Requirement already satisfied: typeguard<3,>=2.13.3 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jaxtyping>=0.2.20->equinox) (2.13.3)\n"
]
}
],
"source": [
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"import matplotlib_inline\n",
"matplotlib_inline.backend_inline.set_matplotlib_formats('svg')\n",
"import seaborn as sns\n",
"sns.set_context(\"paper\")\n",
"sns.set_style(\"ticks\");\n",
"\n",
"!pip show equinox || echo equinox not found. Installing... && pip install equinox 2> /dev/null"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Physics-Informed Neural Networks (PINNs) - Forward Problems\n",
"\n",
"We can use PINNs to solve forward problems in place of traditional numerical methods.\n",
"This is not recommended, as PINNs are not yet as efficient as traditional numerical methods.\n",
"But, it is a good way to learn how to use PINNs and to understand their limitations.\n",
"Throughout this section we will be following closely the methodology of [Wang et al. (2023)](https://arxiv.org/abs/2308.08468).\n",
"\n",
"## The toy problem - Steady-state heat equation\n",
"\n",
"We are going to solve a Poisson's equation with a source term:\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
" -k\\Delta u(x, y) &= f(x, y) \\quad \\text{in} \\quad \\Omega = [0, L_x]\\times [0, L_y], \\\\\n",
" u(x, y) &= 0 \\quad \\text{on} \\quad \\partial \\Omega.\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"## Making an exact solution\n",
"We will use a common trick to construct an exact solution. We will use the following function:\n",
"\n",
"$$\n",
"u(x, y) = u_0 \\sin^2(\\pi x/L_x) \\sin^2(10\\pi y/L_y).\n",
"$$\n",
"\n",
"The boundary conditions are satisfied by construction. The source term is:"
]
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"text/latex": [
"$\\displaystyle \\frac{2 \\pi^{2} k u_{0} \\left(- L_{x}^{2} \\sin^{2}{\\left(\\frac{\\pi x}{L_{x}} \\right)} \\cos{\\left(\\frac{2 \\pi y}{L_{y}} \\right)} - L_{y}^{2} \\sin^{2}{\\left(\\frac{\\pi y}{L_{y}} \\right)} \\cos{\\left(\\frac{2 \\pi x}{L_{x}} \\right)}\\right)}{L_{x}^{2} L_{y}^{2}}$"
],
"text/plain": [
"2*pi**2*k*u_0*(-L_x**2*sin(pi*x/L_x)**2*cos(2*pi*y/L_y) - L_y**2*sin(pi*y/L_y)**2*cos(2*pi*x/L_x))/(L_x**2*L_y**2)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import sympy\n",
"\n",
"xs, ys, ks, u0s, Lxs, Lys = sympy.symbols('x y k u_0 L_x L_y')\n",
"\n",
"us = u0s * sympy.sin(sympy.pi * xs / Lxs) ** 2 * sympy.sin(sympy.pi * ys / Lys) ** 2\n",
"fs = -ks * (sympy.diff(us, xs, 2) + sympy.diff(us, ys, 2))\n",
"sympy.simplify(fs)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is a common trick for constructing exact solutions. Remember it!\n",
"\n",
"Let's use the following parameters:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"u0 = 500 # degrees Kelvin\n",
"k = 10.0 # thermal conductivity in W/mK\n",
"Lx = 0.1 # meters\n",
"Ly = 1.0 # meters"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Enforcing the boundary conditions\n",
"\n",
"We are going to solve this boundary value problem using PINNS.\n",
"Our model will be:\n",
"\n",
"$$\n",
"u(x,y) = x(1-x)y(1-y)\\text{MLP}(x, y),\n",
"$$\n",
"\n",
"where $\\text{MLP}(x, y)$ is a multi-layer perceptron, and $x(1-x)y(1-y)$ is a polynomial that satisfies the boundary conditions.\n",
"This is also a common trick for enforcing boundary conditions in PINNS.\n",
"\n",
"## Multi-layer perceptron\n",
"We will be using a simple multi-layer perceptron to represent the solution of a PDE. The architecture of the network is given by:\n",
"\n",
"$$\n",
"\\text{MLP}(\\mathbf{x}) = z^{(L)},\n",
"$$\n",
"\n",
"where \n",
"\n",
"$$\n",
"z^{(0)} = \\mathbf{x} = (x,y), \\quad z^{(l)} = g^{(l)}(W^{(l)}z^{(l-1)} + b^{(l)}), \\quad l=1,\\ldots,L-1,\n",
"$$\n",
"\n",
"The terms $W^{(l)}$ and $b^{(l)}$ are the weights and biases of the $l$-th layer, and $g^{(l)}$ is the activation function of the $l$-th layer.\n",
"The parameters $\\theta$ are the weights and biases of the network:\n",
"\n",
"$$\n",
"\\theta = \\left\\{\\left(W^{(l)}, b^{(l)}\\right)\\right\\}_{l=1}^L.\n",
"$$\n",
"\n",
"For PINNS applications, it is recommended that:\n",
"+ We use the tanh activation function all layers.\n",
"+ We use 128 to 512 neurons per layer.\n",
"+ We use 3 to 5 layers.\n",
"+ Use the Glorot initialization (this is the default in `equinox`).\n",
"\n",
"Let's make the model:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"import equinox as eqx\n",
"import jax.numpy as jnp\n",
"import jax.random as jrandom\n",
"\n",
"key = jrandom.PRNGKey(0)\n",
"key, subkey = jrandom.split(key)\n",
"# MLP parameters\n",
"width_size = 128\n",
"depth = 4\n",
"mlp = eqx.nn.MLP(2, 1, width_size, depth, jnp.tanh, key=subkey)\n",
"\n",
"# This is the parameterization of the solution that satisfies the boundary conditions\n",
"u_hat = lambda x, y, mlp: x * (Lx - x) * y * (Ly - y) * mlp(jnp.array([x, y]))[0]"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see how it looks like before we train it:"
]
},
{
"cell_type": "code",
"execution_count": 6,
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"