Updated date.
@@ -15,7 +15,7 @@ authors:
affiliations:
- name: Institute for Applied Computational Science, Harvard University, Cambridge, MA, United States
index: 1
-date: 29 October 2019
+date: 31 October 2019
bibliography: paper.bib
---
@@ -71,8 +71,8 @@ The key idea of solving differential equations with ANNs is to reformulate the p
minimize the residual of the differential equations. In a very general sense, a differential equation can be expressed as
$$\mathcal{L}u - f = 0$$
where $\mathcal{L}$ is the differential operator, $u\left(x,t\right)$ is the solution that we wish to find, and $f$ is a known forcing
-function. Note that $\mathcal{L}$ contains any combination of temporal and spatial partial derivatives. Moreover, $x$ is in
-general a vector in three dimensions and $u$ can be a vector of solutions. We denote the output of the neural network as $u_{N}$.
+function. Note that $\mathcal{L}$ contains any combination of temporal and spatial partial derivatives. Moreover, $x$ is, in
+general, a vector in three dimensions and $u$ can be a vector of solutions. We denote the output of the neural network as $u_{N}$.
If $u_{N}$ is a solution to the differential equation, then the residual $$\mathcal{R}\left(u_{N}\right) = \mathcal{L}u_{N} - f $$
will be identically zero. One way to incorporate this into the training process of a neural network is to use the residual
as the loss function. In general, the $L^{2}$ loss of the residual is used. This is the convention that ``NeuroDiffEq`` follows,
