# Application and Evaluation of Surrogate Models for Radiation Source Search
## 3 Surrogate Models  
Surrogate models are used to approximate input-output behavior of complex systems based on limited simulations [17]. For our application, we construct differentiable surrogate models, which approximate the non-differentiable ray-tracing model in Equation (3). These surrogate models are computationally feasible for use with 3D simulation codes or codes incorporating additional physics, such as photon scattering. Whereas there are many methods to create surrogate models, we focus on response surface methods that treat the simulation code as a black box. This is one reason the surrogate modeling techniques we explore are feasible for use with high-fidelity codes. Two surrogate models are based on a polynomial expansion of the physical model evaluations using Legendre polynomials and radial basis functions. We employ discrete projections, which interpolate the data, to evaluate the Legendre polynomial surrogate models. To compare, we consider radial basis function surrogate models constructed using regression and a Legendre polynomial-based surrogate model constructed using sparsity-controlled regression for evaluation of the surrogate models. In addition, we consider a surrogate model based on a multivariate adaptive regression splines (MARS) algorithm. Finally, we demonstrate Gaussian processes and artificial neural networks for surrogate model construction. Our strategy is to construct an individual surrogate model at each of the \(d=10\) randomly chosen detector locations in the urban geometry. To compute the surrogate model response at source locations \((x,y)\in[0,250\ \mathrm{m}]\times[0,180\ \mathrm{m}]\) with intensities \(l\in[5\times 10^{8},5\times 10^{10}\mathrm{Bq}]\), we sample the mean response \(u_{i}(q^{m})\) at \(M=21^{3}=9261\ \mathrm{Gauss}-\mathrm{Legendre}\) training points \(q^{m}\in\Omega=[0,250\ \mathrm{m}]\times[0,180\ \mathrm{m}]\times[5\times 10^{8},5\times 1 0^{10}\ \mathrm{Bq}]\) for each of the \(i=1,...,d\) detector locations. Note that a Bequerel (Bq) is a unit of the activity of nuclear decay in a quantity of radioactive material per second. It takes 4.23 h to compute these 9261 model evaluations for each of the 10 detectors on a computer with a 3.4 GHz Intel core processor and 16 GB of memory. Whereas the computations can be parallelized, we report serial values in Section 5. For each of the \(d\) detectors, we denote the surrogate model by \(u_{i}^{N},\ i=1,\ldots,d\), and relate it to the physical model solution \(u_{i}=\hat{u}_{i}+B\) via the statistical model  
\[u_{i}^{N}(q^{m})=u_{i}(q^{m})+\epsilon_{i}^{m},\ \ m=1,\ldots,M. \tag{8}\]  
The model discrepancies \(\epsilon_{i}^{m}\) quantify unresolved fine-scale behavior incorporated in the ray-tracing model but neglected in the surrogate model. It is reasonable to assume that \(\epsilon_{i}^{m}\) are identically distributed with variance \(\sigma_{0}^{2}\), but one cannot justify the more stringent assumption that \(\epsilon_{i}^{m}\sim\mathcal{N}(0,\sigma_{0}^{2})\). For all but neural networks, we can express surrogate models as  
\[u_{i}^{N}(q,w^{i})=\sum_{j=1}^{N}w_{j}^{j}\Psi_{j}(q)+P_{i}(q), \tag{9}\]  
where \(w^  
behavior of the training data. Additionally, we note that the surrogate model has smoothed the surface represented by the training data, most notably near the singularity at the detector location caused by the \(||r_{i}-r_{S}||\) term in the denominator of Equation (3). To verify each of the \(d\) models, we compare the surrogate and physical models at \(S=500\) randomly generated test points \(q^{s}\in\Omega\). The physical model takes 13.63 min to compute detector responses for the test points. We quantify errors by computing the relative root mean square error (\(rRMSE\))  
\[rRMSE_{i}\equiv\left[\frac{1}{S}\sum_{s=1}^{S}\Big{(}\frac{u_{i}(q^{s})-u_{i} ^{N}(q^{s})}{u_{i}(q^{s})}\Big{)}^{2}\right]^{1/2} \tag{10}\]  
at each detector.