## 1 Introduction  
Computational models using numerical methods tend toward costly performance, in both processing time and hardware resources, as problems become more complex. Typically, such problems take the form of partial differential equations (PDEs), which are used as governing equations to model phenomena in fields such as physics and biology. It is often most convenient to solve these PDEs with numerical solvers, but due to the strain on time and computational resources that comes with numerical methods, surrogate models are used to mitigate the costs. Machine learning (ML) has long been a staple of surrogate modeling and advancements in popular ML libraries for the Python programming language have made ML models, especially deep neural networks (DNNs), easier than ever to implement. DNNs are a type of ML model that, when trained properly, serve as a parameterized solution to mathematical problems. Training a DNN usually requires a large sample pool of data and, in the case of surrogate modeling, DNNs require simulation-driven data collection of the model being surrogated. When a model becomes computationally expensive to exercise, generating sufficient data is less practical. In these situations, DNN models needing less data for convergence become more attractive and one such approach is through implementation of a physics-informed neural network (PINN). PINNs are essentially DNNs, with differences in how the loss function is formulated. Originally restricted to solutions with a trial solution [1], ML advancements have since allowed PINNs to use soft constraints in the loss function to become the unique solution to the PDE, rather than relying on big data to determine relationships. PINNs are able to build a network to solve PDEs in less time, while requiring less data than would be needed for a standard DNN. These effects have been demonstrated on benchmark problems such as [PERSON]'s equation and Schrodinger's equation [2; 3], and on nonlinear diffusivity and Biot's equation [4], as well as on heat transfer problems [5]. Variations of PINNs, called biological informed neural networks (BINNs), have also been used to model the reaction-diffusion equation for sparse real-world data from scratch assay experiments [6]. PINNs, with their smart use of data and recent application to benchmark problems, can potentially offer relief to the computational demand in solving a PDE numerically. The intentions of the following experiments are to use PINNs for solving instances of the diffusion equation, which models how heat diffuses within materials. Its one-dimensional PDE can be represented by the following:  
\[\frac{\partial U}{\partial t}=\alpha\frac{\partial^{2}U}{\partial x^{2}}+Q, \tag{1}\]  
where \(\alpha\) denotes the material's thermal diffusivity and \(U\) represents the solution. This version of the PDE includes a source term \(Q\) representing heat contributed by laser irradiation. Boundary conditions used for solving Equation (1) include insulated and convective boundary conditions. Solving this PDE will result in a function that gives the temperature at a time and depth of the material(s). The open questions addressed in this work are: Can PINNs model the behavior of laser interactions with materials (i.e., can they account for source terms in the PDE)? Can PINN models handle insulated and convective boundary conditions without a source term? Can PINNs handle insulated boundary conditions with a source term? What is the best combination of activation functions for predicting values outside the range of (0,1) for PINNs? Answering these questions would be a critical step in the direction of surrogate modeling with physics-informed ML models. Rapid numerical solvers such as the Python Ablation Code (PAC1D) can approximate a function satisfying Equation (1), so version 4.1.1 of this model will be used as a test bench for prototyping the PINN approach.