This project involves the use of classical neural networks for the computation of the heat equation with Neumann boundary conditions and a gaussian distribution as initial condition.
The universal approximation theorem guarantees the power of neural networks to approximate any function. This means solving partial differential equations (PDEs) too.
In the simplest construction of the network, the solution $u(t,x)$ is given by $u(t,x)\simeq nn(t,x;\theta)$ where $nn(t,x;\theta)$ is the output and $\theta$ are the parameters that minimize best the loss function $$J(\theta)=\dfrac{1}{|\Omega||[0,T]|}\sum_{(t_i,x_i)\in [0,T]\times \Omega}\left(\hat{\mathcal{L}}nn(t_i,x_i;\theta)-s(t_i,x_i)\right)^2+J_{IC}+J_{B}$$ where $\hat{\mathcal{L}}$ is the differential operator and $IC$ and $B$ are the initial and boundary conditions respectively. As an optimization problem, the possiblity of getting stuck into some local minimum is not negligible. An improvement could be the 'weighting' of the loss function, for example $J(\theta)=p_1 J_{\mathcal{L}}+p_2 J_{IC}+p_3 J_{B}$ with $p_1+p_2+p_3=1$ and $p_i\in(0,1)$, in order to force the optimizer to give higher account to some terms of the loss function. Another powerful way to improve the convergence is the construction of the solution in order to automatically satisfies initial and boundary conditions. Such minimization problem is called unconstrained, since the only term to be minimized is the loss of the differential equation. The solution could be written as $$u(t,x)=G(t,x)+D(t,x)nn(t,x;\theta)$$ where $G$ represents the initial and boundary conditions and $D$ is a distance function for $(t,x)\in\times[0,T]\times\Omega$ to $(t=0)\times\partial\Omega$. The ansatz is inspired by the first article below. Theoretically, the complexity of the process is due to the training of $nn(t,x;\theta)$, since $G(t,x)$ and $D(t,x)$ are defined as pre-trained functions with low capacity neural network. The effectiveness of this approach has been demonstrated in numerous articles, particularly when Dirichlet conditions are applied to certain partial differential equations (PDEs). The reason is that Dirichlet conditions are easy to be computed and the distance function could be written analytically as $D(x)=min_{x_0\in\partial\Omega}||x-x_0||$ in case of problems with no temporal dependence.
In this project, we try to solve the heat equation with Neumann conditions. The problem is
$$\partial_t u(t,x)=\alpha\partial^2_x u(t,x)$$$$u(0,x)=f(x)$$$$\partial_x u(t,L)=\partial_x u(t,-L)=0$$ with $(t,x)\in[0,T]\times[-L,L]$. Following the above ansatz, the functions $G$ and $D$ should obey to
$$G(0,x)=f(x),\quad \partial_x G(t,L)=\partial_x G(t,-L)=0,$$$$D(0,x)=0,\quad \partial_x D(t,L)=\partial_x D(t,-L)=0, \quad D(t,L)=D(t,-L)=0.$$ The functions $G$ and $D$ are pre-trained. In particular, the training of $D$ consists of the minimization of $$J_D = \dfrac{1}{|\Omega||[0,T]|}\sum_{(t_i,x_i)\in [0,T]\times[-L,L]}\left(D(t_i,x_i;\theta)-d(t_i,x_i)\right)^2$$ where $d(t,x)=\text{min}(t(x-L)^2,t(x+L)^2)$, which yields to a smooth extension of this distance function. This choice of the distance function ensures that it satisfies the conditions by design. However, empirically, the more conditions there are, the more there is a need for a complex network. For both $G$ and $D$ we use a neural network with 6 hidden layers with 128 neurons each and the relu activation function, 2 input neurons for (t,x) and 1 output neuron with no activation function. We choose $f(x)$ to be a gaussian distribution with mean value $\mu=0$ and variance $\sigma^2=0.1$ while the domain is $[0,T]\times[-L,L]=[0,1]\times[-1,1]$. The optimizer is Adam with a learning rate of $0.01$ and the number of points $t$ and $x$ are $101$ and $201$, equally spaced. Furthermore, $\alpha=1$. Below, there are the plots of $G(t,x)$ for some value of $t$ (including $f(x)$ for $t=0$), the loss of $G$ as a function of the epochs in a semilog graph, $D(t,x)$ for some value of $t$ together with $d(t,x)$ and the loss of $D$ as a function of the epochs in a semilog graph. The convergence has been set to $10^{-6}$ for both.
For the training of $u(t,x)$, instead, we choose $10$ hidden layers with $128$ neurons each, $201$ and $401$ equally spaced $t$ and $x$ points and a threshold of $0.5\times 10^{-6}$. Here, there are the plots of the heatmap of the solution and the loss as a function of the epochs.
This project involves the use of classical neural networks for the computation of the heat equation with Neumann boundary conditions and a gaussian distribution as initial condition.
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