SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Using the Coupled Dipole Approximation (CDA) approach, we compute the extinction and scattering efficiencies  $Q_{ext}$  and  $Q_{sc}$  for plasmonic systems of different scatterer quantities in the final illustration. CDA represents a technique for emulating the optical response of an ensemble of identical, non-magnetic metallic nanoparticles, each with dimensions significantly smaller than the wavelength of light (in this case,  $25\mathrm{nm}$ ). In this work, we define  $Q_{ext}$  and  $Q_{sc}$  as the QoSs. Our proposed method allowed us to construct surrogate models efficiently, incorporating up to three fidelities. We examined particle arrays generated by Vogel spirals, where the interaction of incident waves with particles significantly influences the magnetic field. Consequently, the local extinction field produced by plasmonic arrays is greatly impacted by the number of nanoparticles. The configurations of Vogel spirals, characterized by particle numbers in the set 2, 50, 200, 500, 1000, defined
the parameters for five-fidelity simulations. The parameter space was determined to be  $\lambda \in [200,800]\mathrm{nm}$ ,  $\alpha_{vs} \in [0,2\pi]$  rad, and  $a_{vs} \in (1,1500)$ . These parameters represent the incidence wavelength, the divergence angle, and the scaling factor, respectively. Inputs were chosen using a Sobol sequence. As the number of nanoparticles increases, the computing time required for CDA execution grows exponentially. Hence, the suggested sampling approach leads to substantial reductions in computational costs.
By solving the corresponding linear equation, one can compute the local field  $\mathbf{E}_{loc}(\mathbf{r}_j)$  for each nano-sphere, given  $N$  metallic particles with the same volumetric polarizability  $\alpha (\omega)$  situated at vector coordinates  $\mathbf{r}_i$ . This calculation allows for the determination of the response of a plasmonic array to electromagnetic radiation, as per the solution [61] of the local electric fields,  $\mathbf{E}_{loc}(\mathbf{r}_j)$ .
$$
\mathbf {E} _ {l o c} \left(\mathbf {r} _ {i}\right) = \mathbf {E} _ {0} \left(\mathbf {r} _ {i}\right) - \frac {\alpha k ^ {2}}{\epsilon_ {0}} \sum_ {j = 1, j \neq i} ^ {N} \tilde {\mathbf {G}} _ {i j} \mathbf {E} _ {l o c} \left(\mathbf {r} _ {j}\right), \tag {A.54}
$$
where the incident field is denoted by  $\mathbf{E}_0(\mathbf{r}_i)$ , where  $k$  represents the wave number in the background medium, and  $\epsilon_0$  signifies the dielectric permittivity of vacuum ( $\epsilon_0 = 1$  in the CGS unit system).  $\tilde{\mathbf{G}}ij$  is derived from  $3\times 3$  blocks of the overall  $3N\times 3N$  Green's matrices for the  $i$ th and  $j$ th particles. When  $j = i$ ,  $\tilde{\mathbf{G}}ij$  becomes a zero matrix. Otherwise, it is computed as
$$
\tilde {\mathbf {G}} _ {i j} = \frac {\exp (i k r _ {i j})}{r _ {i j}} \left\{\mathbf {I} - \hat {\mathbf {r}} _ {i j} \hat {\mathbf {r}} _ {i j} ^ {T} - \left[ \frac {1}{i k r _ {i j}} + \frac {1}{(k r _ {i j}) ^ {2}} \left(\mathbf {I} - 3 \hat {\mathbf {r}} _ {i j} \hat {\mathbf {r}} _ {i j} ^ {T}\right) \right] \right\}. \tag {A.55}
$$
Expressed as  $\widehat{\mathbf{r}}ij$ , the unit position vector represents the distance from particle  $j$  to  $i$ , denoted by  $rij = |\mathbf{r}_{ij}|$ . Solving Eq [54] and [55] enables the computation of the overall local fields  $\mathbf{E}_{loc}(\mathbf{r}_i)$ , which in turn determines the scattering and extinction cross-sections. Further information on the numerical approach is available in [62].
Derived from the normalization of the scattering and extinction cross-sections concerning the array's entire projected area  $Q_{ext}$  and  $Q_{sc}$  are considered. Our focus was on the Vogel spiral class of particle arrays, as described in [63].
$$
\rho_ {n} = \sqrt {n} a _ {v s} \quad \text {a n d} \quad \theta_ {n} = n \alpha_ {v s}, \tag {A.56}
$$
In a Vogel spiral array, the  $n$ -th particle's radial distance and polar angle are denoted by  $\rho_{n}$  and  $\theta_{n}$ , respectively. Hence, the unique definition of the Vogel spiral configuration includes the incidence wavelength  $\lambda$ , the divergence angle  $\alpha_{vs}$ , the scaling factor  $a_{vs}$ , and the number of particles  $n$ .
J.4 Detailed Prediction Analysis
Defining the average RMSE field  $\mathbf{y}^{(\mathrm{EF})}$  allows us to thoroughly examine the prediction error.
$$
\mathbf {y} ^ {\left(\mathrm {E F}\right)} = \sqrt {\frac {1}{N} \sum_ {i = 1} ^ {N} \left(\mathbf {y} _ {i} - \tilde {\mathbf {y}} _ {i}\right) ^ {2}}, \tag {A.57}
$$
The square root is taken element-wise, where  $\tilde{\mathbf{y}}_i$  represents the prediction and  $\mathbf{y}_i$  is the true ground value.
The average RMSE field corresponding to the Heat equation, Burger's equation, Poisson's equation, and TopOP problem with a decreasing rate of  $\eta = 0.5$ , lowest-fidelity training samples,  $N^0 = 128$  and 128 testing samples are shown in Fig. (left) for the subset assessment and Fig. (right) for the non-subset assessment. Plasmonic nanoparticle arrays (PNA) have only two output variables and thus we do not show the average RMSE field for it.
For the classic subset assessment in Fig. (left), we can see that clear that ContinuAR outperforms the competitors in all cases with a large margin by showing more blue areas and only tiny red areas. For Heat equation, the error is significantly reduced in most areas except for the bottom area of the domain, where a tiny thin bar of red area is shown. For the Burger's equation, ContinuAR shows some checker board pattern in the error field, which is probability caused by the conditional independence assumption. Nonetheless, the error is significantly reduced in most areas the largest error is also reduced. For the Poisson's equation, different method has it own error patterns. ContinuAR show a more blue area in the left part of the domain, while AR has high-error areas everywhere except the center. For the TopOP problem, ContinuAR show a significantly reduced error by a significant reductions of deep red areas. The AR as the deepest blue ares, indicating its good performance. However, it also has a lar areas of red, indicating its poor performance overall.
For the classic non-subset assessment in fig: errorplot both (right), the overall conclusion is similar to the subset assessment. ContinuAR outperforms the competitors in all cases with a large margin by showing more blue areas and tiny red areas. The overall error pattern for most methods are similar to the subset assessment with subtle difference. For instance, for the Burger's equation, checker board pattern disappear, indicating a successful improvement by using training data across the input domain. The error for Poisson's equation is
Figure 9: Subset Evaluation with  $\eta = 0.5$ : Test negative log-likelihood against number of training samples  $N^0$  for Heat (left), Burger's (middle), and Poisson's (right) equation.
Figure 10: Two-fidelity subset evaluation with  $\eta = 0.5$ : RMSE against the number of training samples  $N^0$  for Heat (left), Burger's (middle), and Poisson's (right) equation.
also significantly reduced particularly on the right part of the domain. TopOP turns out to be the most stable problem as the error patterns for all methods are almost the same as the subset assessment.
K.1 Likelihood Evaluation
Here are some additional results in Figure 9, which shows the negative log-likelihood (without the constant term and thus the result can be negative) of ContinuAR, ResGP, and AR. Surprisingly, the NLL of our method is better than ResGP and AR more significantly than the comparisons in RMSE. We believe that this is because the log-likelihood is more sensitive to the uncertainty of the prediction. Since our method is a joint learning (as discussed in the previous question), it is able to capture the uncertainty as a joint model whereas the other methods treat each fidelity separately.
K.2 Two-fidelity Evaluation
In order to understand the limitation of our method, we follow the classic subset experiment setting with  $\eta = 0.5$  and reduce the number of fidelity from five to two. The results are shown in the Figure 10. In this case, we often see that ContinuAR is almost identical to the classic AR, losing its advantages as an infinite-fidelity fusion method. Also, when the number of training data is scarce, our method does not perform better compared to other baselines. There are certainly other factors that may affect the performance, such as the choice of fidelity and  $\mathrm{B}(\mathrm{x})$ . We will investigate this in future work.
Supplementary References
[60] Tyler E. Bruns and Daniel A. Tortorelli. Topology optimization of non-linear elastic structures and compliant mechanisms. Computer Methods in Applied Mechanics and Engineering, 190(26):3443 - 3459, 2001. ISSN 0045-7825.
[61] Charles-Antoine Guérin, Pierre Mallet, and Anne Sentenac. Effective-medium theory for finite-size ag-gregates. JOSA A, 23(2):349-358, 2006.
[62] Mani Razi, Ren Wang, Yanyan He, Robert M. Kirby, and Luca Dal Negro. Optimization of Large-Scale Vogel Spiral Arrays of Plasmonic Nanoparticles. Plasmonics, 14(1):253-261, February 2019. ISSN 1557-1955, 1557-1963. doi: 10.1007/s11468-018-0799-y.
[63] Aristi Christofi, Felipe A Pinheiro, and Luca Dal Negro. Probing scattering resonances of vogels spirals with the greens matrix spectral method. Optics letters, 41(9):1933-1936, 2016.