## 2 Methods
### PAC1D and Data Generation  
PAC1D was designed for modeling laser-tissue interaction and numerically solves the 1-D Advection-Diffusion-Reaction equation with a source term representing thermal energy contributed by a laser source. An example of a three-layer skin model with a laser directed at its surface can be seen in Figure 3. For the testing performed in this paper, advection and reaction terms in the governing equation did not contribute to the solution of the PDE simplifying to Equation (1). Features of the model used for these experiments were: defining boundary conditions, defining material-specific optical and mechanical properties, and numerically solving for the domain temperature \(U(x,t)\), where \(x\in[0,x_{L}]\) and \(t\in[0,T_{\tau}]\). For the purposes of surrogate modeling, data obtained from PAC1D are considered to be the true solution values. Figure 2: Automatic differentiation component of the PINN model, taking partial derivatives of the parameters in the network with respect to \(x\) and \(t\) from collocation points. Initial and boundary conditions are essential to finding a unique solution to a PDE. There are several types of boundary conditions to consider, and two of particular interest in this study are insulated and convection. Under insulated boundary conditions, thermal gradients at the boundaries are set to zero,  
\[\frac{\partial U}{\partial x}(0,t)=\frac{\partial U}{\partial x}(x_{L},t)=0 \tag{7}\]  
where \(t\in[0,T_{\tau}]\). Convection in 1-D allows for a thermal flux to exist at the boundaries of the system,  
\[H=h(T_{a}-T_{s}), \tag{8}\]  
and its rate depends on a material's convective coefficient, \(h\), as well as the differences between ambient and surface temperatures, \(T_{a}\) and \(T_{s}\), respectively. A three-layer model of skin was represented in the 1-D domain by defining layer-specific thicknesses and material properties. The PDE solution relies not only on initial and boundary conditions but also on the physical properties of the materials. Hence, in the data that PAC1D generated, the necessary material properties were extracted for use in the training process. The source laser energy deposition is tracked as a dose irradiance and depends on each layer's optical properties in the domain. Dose tends to vary throughout the domain of each material due to photon scattering and absorption by each material. These dose distributions are relevant in the training process when the source term is incorporated into the PDE and solved. Generating PAC1D output data required defining the parameters of the laser exposure, thermal boundary conditions, material properties, material geometry, total simulation time, spatial resolution, and temporal resolution. The laser frequency was controlled by adjusting the tissue-specific absorption and reduced scattering coefficients. Time and depth values were chosen arbitrarily but lead to convergence of the PDE solution. Dose and temperature values derived from PAC1D shared the same spatial and temporal resolution. Collocation points were generated using the LHS method in non-source experiments, and the random choice function from the Numpy library [19] in Python, version 1.19.5, for source experiments. Because there are no dose values associated with the non-source experiments the LHS method could be used, as these values did not have to be consistent with any depth and dose values obtained from PAC1D simulation. This means that collocation points were not restricted to being the exact points used in the PAC1D simulation. With the source experiment, however, collocation points must have a corresponding dose value for each. Because dose values for the source experiment are taken directly from PAC1D, and because PAC1D only stores dose values for each of its spatial points from the simulation, collocation points required a random sampling from the \(x\) and \(t\) data values retrieved from PAC1D. This would result in a more concentrated sampling at the earlier layers of the model.