\(a(x)\) from spare measurement
Regression loss: \[ L_\text{reg}(\theta)= \sum_{j} [ u(t_j, x_j;\theta) - u(t_j, x_j) ]^2. \]
Residual loss: \[ L_\text{res}(\theta) \propto \int_{0}^{1} \int_{-1}^{1} \left[ \frac{\partial}{\partial t}\left(\frac{u_t(t,x;\theta)}{u_{xx}(t,x;\theta)} \right) \right]^2 \mathrm{d}x \mathrm{d}t. \]
This takes advantage of the form of the PDE: \[ a(x) = \frac{u_t(t,x)}{u_{xx}(t,x)} \]