Upsampling Monte Carlo Reactor Simulation Tallies in Depleted Sodium-Cooled Fast Reactor Assemblies Using a Convolutional Neural Network

Abstract

The computational demand of neutron Monte Carlo transport simulations can increase rapidly with the spatial and energy resolution of tallied physical quantities. Convolutional neural networks have been used to increase the resolution of Monte Carlo simulations of light water reactor assemblies while preserving accuracy with negligible additional computational cost. Here, we show that a convolutional neural network can also be used to upsample tally results from Monte Carlo simulations of sodium-cooled fast reactor assemblies, thereby extending the applicability beyond thermal systems. The convolutional neural network model is trained using neutron flux tallies from 300 procedurally generated nuclear reactor assemblies simulated using OpenMC. Validation and test datasets included 16 simulations of procedurally generated assemblies, and a realistic simulation of a European sodium-cooled fast reactor assembly was included in the test dataset. We show the residuals between the high-resolution flux tallies predicted by the neural network and high-resolution Monte Carlo tallies on relative and absolute bases. The network can upsample tallies from simulations of fast reactor assemblies with diverse and heterogeneous materials and geometries by a factor of two in each spatial and energy dimension. The network’s predictions are within the statistical uncertainty of the Monte Carlo tallies in almost all cases. This includes test assemblies for which burnup values and geometric parameters were well outside the ranges of those in assemblies used to train the network.

1. Introduction

The significant computational demand associated with high-resolution Monte Carlo simulations of nuclear reactor cores has motivated the development of CPU and memory-efficient methods to increase the simulations’ fidelity. Monte Carlo codes, such as Serpent [1], OpenMC [2], and MCNP [3], implement a variety of such methods, including Functional Expansion Tallies (FETs) [4] and domain decomposition. FETs can increase the resolution of neutron flux tallies by expanding the spatial and angle-dependent neutron flux using orthogonal functions. Methods based on domain decomposition divide the workload of a single compute node among several parallel nodes. This method is useful when the simulation is too large to fit in memory and has been implemented in Serpent, as well as in a developmental version of OpenMC [5,6]. Disjoint tallies have a basis in compressed sensing methods and have also been shown to yield significant memory reductions in neutron Monte Carlo simulations [7,8].

Machine learning approaches have also been used to increase neutron flux tallies’ spatial and energy resolution without any significant increase in CPU or memory requirements. Convolutional neural networks (CNNs) were used in [9,10] to upsample Monte Carlo neutron flux tallies in simulations of light water reactor (LWR) assemblies. The low-resolution tallies were calculated at a spatial resolution of 64 × 64 pixels and with eight neutron energy groups and were upsampled to 128 × 128 pixels and 16 groups. Although the CNNs were able to upsample LWR tallies accurately over a diverse range of assembly geometries, fuel enrichment, and burnup levels, their applicability to fast reactors was not established. Here, we show that a CNN can be used to upsample neutron flux tallies from Monte Carlo simulations of sodium-cooled fast reactor (SFR) assemblies. We constrained the training data to heterogeneous hexagonal assemblies with one to five rings of pins and with fuel burned up to 180 MWd/kgIHM. Fuel isotopics in the training data were taken from depletion simulations of LWR assemblies fueled with uranium dioxide enriched to 1.6%, as well as fresh fuel. The fresh-fueled assemblies in the training data included uranium dioxide (UOX) and pure uranium metal enriched to 11%. To increase the diversity of the training data, we also included samples with thermal spectra and training samples with boron-shielded fuel and no coolant that produced very hard spectra. The validation data consisted of SFR assemblies with 10 and 11 rings of pins and fuel with burnup levels of up to 180 MWd/kgIHM. The test data included SFR assemblies with 10 and 11 rings of pins and fuel with burnup levels of up to 400 MWd/kgIHM. Fuel isotopics in the test and validation data originate from depletion simulations of LWR assemblies with UOX enriched to 1.6% and 19.9%, as well as fresh fuel. Fuel in the testing and validation data included UOX, mixed-oxide (MOX), and uranium–plutonium–zirconium (UPuZr) alloys. All assemblies in the training and validation datasets were generated by randomly selecting geometry, temperatures, and materials using Latin hypercube sampling. This was also done to generate the test data, although this dataset also included a realistic assembly simulation of the European sodium-cooled fast reactor (ESFR) [11].

The CNN topology was the same as in [10], which included three residual blocks and a 3 × 3 kernel in all convolutional layers. The CNN in this work, however, featured 256 filters in each convolutional layer and exponential linear unit (ELU) activations. Inputs to the CNN and ground truth quantities used to calculate the loss were low- and high-resolution flux tallies, respectively, computed using OpenMC. The accuracy of the CNN was shown by comparing the residuals between the predictions and ground truth with the standard deviation in the ground truth tallies computed using OpenMC. We also calculated the fraction of predictions made by the CNN that lie outside one and two standard deviations of the ground truth. We found that a residual CNN can increase the resolution of 2-D neutron flux tallies in simulated SFR assemblies by a factor of two in the spatial and energy dimensions. The accuracy of the upsampled neutron flux tallies calculated using the CNN was comparable to the uncertainty in the tallies computed using high-resolution Monte Carlo simulations.

2. Methods

2.1. Data Generation

All datasets consisted of neutron flux mesh tallies and their uncertainties at low and high resolutions generated using the OpenMC Monte Carlo code [2] and ENDF/B-VIII.0 [12] nuclear data. The spatial resolution of the low- and high-resolution tallies was 0.1 × 0.1 cm and 0.05 × 0.05 cm, respectively. The energy grids of the low- and high-resolution tallies consisted of 8 and 16 groups, respectively, with group boundaries in equal logarithmic spacing between 100 eV and 20 MeV. Criticality eigenvalue simulations used 50 inactive cycles, 100 active cycles, and 100,000 source neutrons per cycle. Input decks for OpenMC were created using the OpenMC Python API and an in-house Python code [13]. The Python code used Latin hypercube sampling to select the geometry and materials of the assembly and individual fuel pins according to specified ranges and choices. Individual fuel pin parameters that could be randomly selected were fuel radius, temperature and burnup level, and cladding thickness. The sampling method could also randomly select B₄C or sodium for the contents of the pin. The randomly chosen assembly parameters included coolant temperature, duct thickness, and the number of rings of pins. The ranges of all geometric parameters used in the generation of training, validation, and testing datasets are given in

Table 1. Ranges of geometric parameters in assembly simulations.

Parameter	Range (Procedurally Generated Assemblies)	Value (ESFR Test Assembly)
Pin Diameter a	0.5 to 1.0	0.4715 cm
Pin Pitch [cm]	0.51 to 1.32	1.073 cm
Clad Thickness a	0.0633 to 0.0942	0.5 mm
Duct Thickness [cm]	0 to 0.24	0.45 cm
Coolant Temperature [K]	350 to 800	743
Fuel Temperature [K]	500 to 1428	1624
Assembly Rings	2 to 5 (training) 10 to 11 (validation and testing)	10

^a Pin diameter and clad thickness are expressed as a fraction of the pin pitch for procedurally generated assemblies.

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The training datasets were compiled from simulations of assemblies with 2 to 5 rings of pins and fuel burnup levels ranging from 0 to 180 MWd/kgIHM. Validation and testing datasets were compiled from simulations of assemblies with 10 or 11 rings of pins and fuel burnup levels of up to 400 MWd/kgIHM. These values in the validation and test datasets were chosen to show that the CNN could make accurate predictions in assemblies with geometry and burnup well beyond those in the training data. The test dataset also contained a simulation of an ESFR assembly to show that the CNN could accurately upsample simulations of assembly models with realistic geometries and materials. The geometry of the selected representative sample assemblies from the training data and validation/test data are shown in Figure 1.

The assemblies in all datasets contained a mix of UPuZr, UOX, MOX, and pure metal uranium fuel.

Table 2. Datasets used in training, validation, and testing.

Dataset	Number of Samples	Data Type	Fuel Type	Plutonium Origin	Spectrum	Burnup Range [MWd/kgIHM]	Fraction of B4C, Empty Pin Positions
A	50	Training	UPuZr a	1.6% UOX b LWR	SFR	0 to 180	14%, 14%
B	50	Training	UOX b	N/A (Fresh 11% UOX b in pincell sim)	SFR (Softened, OX Fuel)	0 to 180	14%, 14%
C	50	Training	UOX b	N/A (Fresh UOX b)	SFR (Softened, OX Fuel)	0	[0% or 14%], [14% or 17%]
D	50	Training	UPuZr a	1.6% UOX b LWR	LWR d	0	[0% or 14%], [0% or 14% or 17%]
E	50	Training	UPuZr a	1.6% UOX b LWR	Hard e	0	[0% or 14% or 17%], [0% or 14% or 17%]
F	50	Training	U b	N/A (Fresh 11% U)	Hard f	0	0%, 0%
G	2	Validation	UPuZr a	1.6% UOX b LWR	SFR	0 to 180	14%, 14%
H	2	Validation	UPuZr a	19.9% UOX b LWR	SFR	0 to 180	14%, 14%
I	2	Validation	UOX b	N/A (Fresh UOX b)	SFR (Softened, OX Fuel)	0 to 180	14%, 14%
J	2	Validation	MOX b,c	1.6% UOX b LWR	SFR (Softened, OX Fuel)	0 to 180	14%, 14%
K	1	Test	MOX b,c	N/A (Fresh MOX b)	SFR (Softened, OX Fuel)	0	0%, 0%
L	2	Test	UPuZr a	1.6% UOXb LWR	SFR	220 to 400	14%, 14%
M	2	Test	UPuZr a	19.9% UOX b LWR	SFR	220 to 400	14%, 14%
N	2	Test	UOX b	N/A (Fresh UOX b)	SFR (Softened, OX Fuel)	220 to 400	14%, 14%
O	2	Test	MOX b,c	1.6% UOX b LWR	SFR (Softened, OX Fuel)	220 to 400	14%, 14%

^a Fuel contained 10% Zr by weight. The fractions of U and Pu were 89% and 11% by weight, respectively. A 75% smear density was used in all UPuZr-fueled pins. ^b U was enriched to 11% using atomic fraction. ^c The fractions of U and Pu were 89% and 11% by weight, respectively. ^d LWR spectrum was achieved by replacing Na coolant with water. ^e Hard spectrum was achieved using a combination of fuel pins clad with B4C and assemblies with no coolant. ^f Hard spectrum was achieved using a combination of fuel pins clad with B4C and assemblies with no coolant. All fuel was pure U metal.

summarizes the main characteristics of the training (A-F), validation (G-J), and test (K-O) datasets. The fuel isotopics in all datasets containing depleted assemblies (A, B, G-J, L-O) were determined using SFR pincell burnup simulations performed in OpenMC. Figure 2 shows the geometry used in the SFR pincell burnup simulations. Training datasets A, E, and D contained either UPuZr or MOX fuel with plutonium isotopics originating from UOX enriched to 1.6% and burned to 17.44 MWd/kgIHM in an LWR. The LWR depletion simulations had been performed previously in [10]. Training dataset B contained UOX enriched to 11% and burned up to 180 MWd/kgIHM in the SFR pincell depletion simulations. Training datasets C and F contained fresh UOX and fresh pure U, respectively, enriched to 11% in both cases. Figure 3 shows a flow diagram indicating the origin of all fuel types used in each training dataset.

Validation datasets G, H, and J contained either UPuZr or MOX fuel with plutonium isotopics originating from UOX burned in an LWR. In datasets G and J, the UOX was enriched to 1.6% and burned to 17.44 MWd/kgIHM. In the case of dataset H, the UOX was enriched to 19.9% and burned to 216.91 MWd/kgIHM. Validation dataset I contained UOX enriched to 11%. All MOX, UPuZr, and UOX fuels were then burned in SFR pincell depletion simulations up to 180 MWd/kgIHM before being used in assembly simulations. Figure 4 shows a flow diagram indicating the origin of all fuel types used in each validation dataset.

Test datasets M, N, and O contained either UPuZr or MOX fuel with plutonium isotopics originating from UOX burned in an LWR. In datasets L and O, the UOX was enriched to 1.6% and burned to 17.44 MWd/kgIHM. In dataset M, the UOX was enriched to 19.9% and burned to 216.91 MWd/kgIHM. Test dataset N contained UOX enriched to 11%. All MOX, UPuZr, and UOX fuels in test datasets L, M, N, and O were then burned in SFR pincell depletion simulations from 220 to 400 MWd/kgIHM before being used in assembly simulations. Test dataset K contained MOX fuel with isotopics from the ESFR reactor [11]. Figure 5 shows a flow diagram indicating the origin of all fuel types used in each test dataset.

The plutonium isotopics of the fresh SFR fuels containing plutonium are shown in

Table 3. Plutonium isotopics from SFR fuel containing plutonium. Values are expressed as a percentage of total plutonium.

	Dataset, Pu Origin
Pu Isotope	A, D, E, G, J, L, O 1.6% UOX LWR [Atom %]	H, M 19.9% UOX LWR [Atom %]	K Pu from [11] [Weight %]
236Pu	1.9 × 10−9%	6.3 × 10−7%	0%
237Pu	7.2 × 10−7%	8.5 × 10−6%	0%
238Pu	0.59%	21.47%	3.6%
239Pu	60.0%	35.8%	47.7%
240Pu	23.8%	17.9%	29.9%
241Pu	11.9%	14.0%	8.3%
242Pu	3.7%	10.9%	10.5%
243Pu	0.0014%	0.0024%	0%

for each relevant dataset. The plutonium isotopics of each fuel type that was depleted in SFR pincell simulations are shown in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7 and Figure A8 in Appendix A.

2.2. Convolutional Neural Network

The CNN developed in this work was a residual network [14] similar to the models developed in [9,10]. Residual networks are characterized by the inclusion of skip connections between shallower and deeper layers that bypass layers in between. Residual networks are used to reduce the impact of vanishing gradients during training [15]. A simple, manually guided optimization was conducted where the activation function and number of filters in each convolutional layer of the CNN in [10] were varied. It was found that using ELU activations in place of rectified linear units (ReLUs) and increasing the number of filters from 128 to 256 improved the validation loss. The CNN was developed using the TensorFlow [16] library’s Keras API and was trained for 2500 epochs using the Adam optimization algorithm. Training proceeded with a batch size of 100 and an initial learning rate of 10⁻⁴. The learning rate was reduced to 2 × 10⁻⁵, 1 × 10⁻⁵, and 2 × 10⁻⁶ after 4000, 8000, and 16,000 training steps, respectively. The mean absolute error (MAE) loss metric was used as a basis for updating the weights and biases of the network during training. The validation dataset was used to compute a validation MAE loss score independent of the training loss. The weights and biases of the CNN that achieved the lowest validation loss score over 2500 epochs were stored and used to generate results in this work.

Because procedurally generated assemblies varied in size, the grids on which the mesh tallies were recorded also varied in their number of horizontal and vertical elements. To allow training to be conducted in a batch mode using stochastic gradient descent, each training sample’s input low-resolution tallies were segmented into equal-size non-overlapping 20 × 20 × 8 grids. The ground-truth high-resolution tallies were segmented into corresponding 40 × 40 × 16 grids. The training and inference pipeline included the data pre- and post-processing steps that scaled the input low-resolution data and output high-resolution prediction. All tally values in each sample were first multiplied by the total number of elements in that sample’s mesh. Again, this was done because generated assemblies varied in size, causing the recorded flux per mesh element in larger assemblies to be reduced compared with smaller assemblies. In a second scaling step, the largest flux value was sought for each energy group across the entire set of training data. These flux values were found in both the low- and high-resolution tallies in the training dataset. The input low-resolution tallies and output upsampled tallies were then divided by the largest flux values on a groupwise basis. The largest flux values in the training data were stored for use during inference on unseen validation and test data, where the high-resolution tallies are unknown to the CNN.

3. Results and Discussion

Figure 6 shows the low- and high-resolution neutron flux mesh tallies and relative error from the simulation corresponding to dataset K, as well as the upsampled CNN prediction. Dataset K (Table 2) corresponds to the simulation of the ESFR assembly. In this figure, the mesh tally fluxes are summed in the energy dimension. The relative error was calculated by summing absolute errors in quadrature along the energy dimension and dividing them by the flux. The upsampling prediction in Figure 6c compares well visually with the high-resolution OpenMC tally in Figure 6b. Figure 6d shows that the relative Monte Carlo uncertainty computed by OpenMC is close to 0.5% across the entire assembly.

Figure 7 compares the residuals between the high-resolution OpenMC tally and the upsampled CNN prediction for dataset K as a function of neutron energy on relative and absolute bases. The mean residuals and errors are shown for each energy group, which is averaged across all spatial locations. The results in Figure 7 show that CNN’s prediction residual significantly exceeds the relative error computed using OpenMC only in the first and last energy groups. In all other energy groups, the CNN’s predictions were either within or very close to the Monte Carlo errors on a relative basis. This was the case for all energy groups when results were computed on an absolute basis. The figure also shows that CNN’s predictions only had a strong bias in the first and last energy groups. The neutron flux in these groups, however, was extremely low relative to that of other groups. Monte Carlo uncertainties scaled with the square root of the number of samples, which amplified the relative uncertainty in the energy groups with low statistics. The average neutron flux for dataset K is shown in the Appendix A, Figure A9.

The number of CNN predictions worse than the Monte Carlo uncertainties is a useful metric for additional insight into its prediction accuracy. Figure 8 shows the fraction of CNN predictions outside one and two standard deviations of the Monte Carlo calculation for dataset K. CNN’s predictions were outside one and two standard deviations in 33.6% and 7.6% of cases overall, respectively. These numbers align closely with what would be expected, assuming that errors are normally distributed. By using this metric, the prediction accuracy was reduced at very low and high energies where the flux was low. The results shown in Figure 7 and Figure 8 are tabulated in the Appendix,

Table A1. Tabulated data for Figure 7 and Figure 8, dataset K. Lowest error for each group is in bold.

Upper Energy Bin [eV]	MC Relative Error [%]	CNN Relative Residual [%]	CNN Relative Residual Bias [%]	MC Absolute Error a	CNN Absolute Residual a	Fraction of Predictions Worse Than MC Uncertainty [%]
						1σ	2σ
2.14 × 102	24.38	34.85	29.56	2.98 × 10−8	3.64 × 10−8	52.88	24.25
4.60 × 102	10.06	6.76	1.15	7.43 × 10−8	4.91 × 10−8	23.14	2.27
9.86 × 102	5.14	4.44	2.10	1.47 × 10−7	1.24 × 10−7	35.05	7.02
2.11 × 103	3.26	2.39	−1.24	2.34 × 10−7	1.72 × 10−7	28.07	2.67
4.53 × 103	3.86	3.22	0.87	1.99 × 10−7	1.65 × 10−7	33.72	5.94
9.72 × 103	2.37	1.62	−0.45	3.23 × 10−7	2.22 × 10−7	24.53	2.01
2.09 × 104	1.67	1.68	1.21	4.52 × 10−7	4.49 × 10−7	42.27	11.65
4.47 × 104	1.49	1.35	−0.91	5.12 × 10−7	4.70 × 10−7	38.21	8.11
9.59 × 104	1.31	1.04	0.43	5.77 × 10−7	4.53 × 10−7	31.16	4.59
2.06 × 105	1.22	0.90	0.14	6.17 × 10−7	4.50 × 10−7	27.64	3.17
4.41 × 105	1.32	1.07	0.20	5.70 × 10−7	4.60 × 10−7	32.39	5.18
9.46 × 105	1.40	1.14	−0.04	5.32 × 10−7	4.32 × 10−7	32.84	5.08
2.03 × 106	2.06	1.53	−0.65	3.59 × 10−7	2.68 × 10−7	28.48	3.10
4.35 × 106	2.61	2.13	0.85	2.82 × 10−7	2.28 × 10−7	32.68	5.56
9.33 × 106	5.52	3.14	−0.05	1.32 × 10−7	7.47 × 10−8	16.20	0.64
2.00 × 107	31.57	53.90	50.44	2.16 × 10−8	2.83 × 10−8	58.41	29.68

^a Units are per starting source particle.

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The CNN prediction residuals and Monte Carlo uncertainties are compared for the test dataset L in Figure 9. This dataset comprised procedurally generated assemblies with 10 or 11 rings of pins whose fuel’s plutonium isotopes were taken from discharged LWR fuel enriched to 1.6%. In the assemblies of dataset L, the fuel form was UPuZr with burnup values between 220 and 400 MWd/kgIHM, burned in a fast spectrum. Figure 10 shows the fraction of CNN predictions outside one and two Monte Carlo standard deviations for dataset L. On a relative basis, the mean residuals in the CNN’s predictions were close to or below the Monte Carlo uncertainty, except for the low and high energy groups. Again, these energies correspond to very low neutron fluxes. The prediction bias in either direction was also small except in the groups with low neutron flux values. Unlike dataset K, in dataset L, the mean values of the CNN’s residuals were larger than Monte Carlo uncertainties around 200 to 950 keV. On a relative basis, however, the residuals in this region were 1.45% at worst. The average neutron flux for dataset L is shown in the Appendix A, Figure A10.

Figure 10 shows the fraction of CNN predictions outside one and two Monte Carlo standard deviations for dataset L. CNN’s predictions were outside one and two standard deviations in 37.6% and 9.8% of cases overall, respectively. The results shown in Figure 9 and Figure 10 are tabulated in the Appendix A,

Table A2. Tabulated data for Figure 9 and Figure 10, dataset L. Lowest error for each group in bold.

Upper Energy Bin [eV]	MC Relative Error [%]	CNN Relative Residual [%]	CNN Relative Residual Bias [%]	MC Absolute Error a	CNN Absolute Residual a	Fraction of Predictions Worse Than MC Uncertainty [%]
						1σ	2σ
2.14 × 102	32.55	43.74	−8.21	2.48 × 10−8	3.08 × 10−8	54.07	18.94
4.60 × 102	10.73	7.68	2.65	7.54 × 10−8	5.27 × 10−8	25.88	3.39
9.86 × 102	4.25	3.74	1.78	1.87 × 10−7	1.68 × 10−7	37.05	7.95
2.11 × 103	2.59	1.79	−0.49	3.06 × 10−7	2.17 × 10−7	25.55	2.23
4.53 × 103	3.68	3.51	−2.22	2.17 × 10−7	2.08 × 10−7	41.35	8.98
9.72 × 103	1.93	1.37	0.74	4.07 × 10−7	2.88 × 10−7	26.48	2.78
2.09 × 104	1.26	1.05	−0.56	6.15 × 10−7	5.14 × 10−7	34.30	5.45
4.47 × 104	1.08	0.83	0.36	7.14 × 10−7	5.41 × 10−7	29.62	3.90
9.59 × 104	0.94	0.73	−0.31	8.21 × 10−7	6.42 × 10−7	30.86	4.13
2.06 × 105	0.84	0.62	0.22	9.12 × 10−7	6.73 × 10−7	28.22	3.35
4.41 × 105	0.85	1.00	0.90	8.99 × 10−7	1.06 × 10−6	54.15	15.94
9.46 × 105	1.01	1.45	−1.39	7.58 × 10−7	1.11 × 10−6	67.00	26.27
2.03 × 106	1.38	0.93	0.02	5.51 × 10−7	3.71 × 10−7	23.74	1.93
4.35 × 106	2.10	1.67	−0.03	3.57 × 10−7	2.85 × 10−7	31.72	4.75
9.33 × 106	4.45	2.52	0.07	1.66 × 10−7	9.37 × 10−8	16.06	0.64
2.00 × 107	24.81	53.11	52.09	2.84 × 10−8	5.25 × 10−8	75.21	46.35

^a Units are per starting source particle.

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Figure 11 shows a comparison between the CNN prediction residuals and Monte Carlo uncertainties for test dataset M. This dataset comprised procedurally generated assemblies with 10 or 11 rings of pins whose fuel’s plutonium isotopes were taken from discharged LWR fuel enriched to 19.9%. In the assemblies of dataset M, the fuel form was UPuZr with burnup values between 220 and 400 MWd/kgIHM, burned in a fast spectrum. The accuracy of the CNN’s predictions, as well as the prediction bias, on dataset M varied by energy in a similar manner to dataset L. Although the absolute residuals between 200 and 950 keV were larger than Monte Carlo uncertainty, on a relative basis, this corresponded to 1.64% at worst. The average neutron flux for dataset M is shown in the Appendix A, Figure A11.

Figure 12 shows the fraction of CNN predictions outside one and two Monte Carlo standard deviations for dataset M. CNN’s predictions were outside one and two standard deviations in 35.4% and 8.9% of cases overall, respectively. The results shown in Figure 11 and Figure 12 are tabulated in the Appendix A,

Table A3. Tabulated data for Figure 11 and Figure 12, dataset M. Lowest error for each group in bold.

Upper Energy Bin [eV]	MC Relative Error [%]	CNN Relative Residual [%]	CNN Relative Residual Bias [%]	MC Absolute Error a	CNN Absolute Residual a	Fraction of Predictions Worse Than MC Uncertainty [%]
						1σ	2σ
2.14 × 102	59.16	282.61	196.20	1.91 × 10−8	2.41 × 10−8	53.45	24.56
4.60 × 102	17.95	13.11	1.37	5.68 × 10−8	4.04 × 10−8	26.42	3.37
9.86 × 102	6.48	5.37	−0.07	1.44 × 10−7	1.32 × 10−7	35.39	6.49
2.11 × 103	3.81	2.62	0.47	2.37 × 10−7	1.72 × 10−7	25.42	2.55
4.53 × 103	5.43	4.89	−2.06	1.65 × 10−7	1.65 × 10−7	39.97	8.49
9.72 × 103	2.71	1.85	0.73	3.19 × 10−7	2.30 × 10−7	25.27	2.57
2.09 × 104	1.76	1.36	−0.50	4.84 × 10−7	3.88 × 10−7	31.00	4.16
4.47 × 104	1.49	1.06	0.27	5.67 × 10−7	4.14 × 10−7	26.87	2.99
9.59 × 104	1.28	1.00	−0.36	6.52 × 10−7	5.12 × 10−7	30.85	4.12
2.06 × 105	1.14	0.85	0.27	7.26 × 10−7	5.37 × 10−7	28.12	3.36
4.41 × 105	1.16	1.20	1.00	7.14 × 10−7	8.15 × 10−7	48.10	12.96
9.46 × 105	1.37	1.64	−1.46	5.98 × 10−7	8.17 × 10−7	56.89	19.07
2.03 × 106	1.87	1.27	−0.01	4.35 × 10−7	2.97 × 10−7	24.24	2.05
4.35 × 106	2.80	2.24	0.16	2.85 × 10−7	2.30 × 10−7	32.08	4.92
9.33 × 106	5.86	3.33	0.09	1.33 × 10−7	7.54 × 10−8	16.13	0.68
2.00 × 107	32.88	66.87	64.30	2.27 × 10−8	3.94 × 10−8	66.71	39.41

^a Units are per starting source particle.

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The CNN prediction accuracy for datasets N and O was extremely close to or below the Monte Carlo uncertainties in almost all energy groups, as shown in Figure 13. The sole exception to this was at the highest energy group and only on a relative basis. Because of this, Figure 13 represents the combined statistics of datasets N and O. Prediction biases were also small and, overall, did not show a preference in direction, except for the lowest and highest energy groups. Similarly, Figure 14 shows the fraction of CNN predictions outside one and two Monte Carlo standard deviations for datasets N and O, again using combined statistics for both datasets. CNN’s predictions were outside one and two standard deviations in 33.0% and 6.8% of cases overall, respectively. The results shown in Figure 13 and Figure 14 are tabulated in the Appendix A,

Table A4. Tabulated data for Figs. 13,14, datasets N and O. Lowest error for each group in bold.

Upper Energy Bin [eV]	MC Relative Error [%]	CNN Relative Residual [%]	CNN Relative Residual Bias [%]	MC Absolute Error a	CNN Absolute Residual a	Fraction of Predictions Worse Than MC Uncertainty [%]
						1σ	2σ
2.14 × 102	13.53	13.32	−7.23	6.92 × 10−8	6.47 × 10−8	42.66	9.18
4.60 × 102	5.82	4.03	1.66	1.50 × 10−7	1.01 × 10−7	24.44	2.63
9.86 × 102	2.89	2.36	1.01	2.88 × 10−7	2.30 × 10−7	32.45	5.41
2.11 × 103	2.01	1.39	−0.44	4.09 × 10−7	2.85 × 10−7	25.24	2.19
4.53 × 103	2.87	2.35	−0.42	2.86 × 10−7	2.42 × 10−7	33.87	5.55
9.72 × 103	1.61	1.06	0.25	4.98 × 10−7	3.35 × 10−7	23.26	1.91
2.09 × 104	1.15	0.86	−0.10	6.91 × 10−7	5.19 × 10−7	28.84	3.49
4.47 × 104	1.05	0.75	0.05	7.58 × 10−7	5.48 × 10−7	26.92	2.90
9.59 × 104	0.93	0.68	0.02	8.47 × 10−7	6.22 × 10−7	27.77	3.16
2.06 × 105	0.86	0.62	0.05	9.12 × 10−7	6.55 × 10−7	26.70	2.83
4.41 × 105	0.92	0.81	0.54	8.57 × 10−7	7.56 × 10−7	37.07	7.01
9.46 × 105	1.00	0.93	−0.65	7.76 × 10−7	7.22 × 10−7	39.79	7.94
2.03 × 106	1.35	1.08	−0.66	5.72 × 10−7	4.52 × 10−7	32.24	4.24
4.35 × 106	1.92	1.96	1.39	3.94 × 10−7	3.91 × 10−7	43.78	11.59
9.33 × 106	4.19	2.38	0.21	1.77 × 10−7	1.01 × 10−7	16.26	0.70
2.00 × 107	23.77	43.03	41.12	2.99 × 10−8	4.94 × 10−8	66.78	37.54

^a Units are per starting source particle.

. The average neutron flux for datasets N and O is shown in the Appendix A, Figure A12.

The predictions made by the CNN on unseen test data were accurate on a relative basis, except in energy groups where the neutron flux was close to zero, as shown in Figure 7, Figure 9, Figure 11, and Figure 13. This was also the case on an absolute basis, with the only exceptions being in the 200–950 keV region in high burnup procedurally generated test assemblies fueled with UPuZr, as shown in Figure 9, Figure 10, Figure 11 and Figure 12. Even in these cases, the difference between the CNN predictions’ residuals, 1.64%, and the Monte Carlo uncertainty, 1.37%, was extremely small compared with the neutron flux. In the MOX or UOX test assemblies burned up to 400 MWd/kgIHM, however, the CNN residuals remained smaller than the Monte Carlo uncertainties in the 200–950 keV region, as shown in Figure 7 and Figure 13. A likely explanation for this difference is that the fissile plutonium content in UPuZr fuel is lower than that in the MOX or UOX fuel at high levels of burnup. Even in the high-burnup MOX and UOX samples (Datasets N, O, Figure 13), the predictive accuracy of the CNN in the 200–950 keV region was slightly worse compared with the ESFR assembly predictions, as shown in Figure 7. In the ESFR assembly, the plutonium content of the fuel was high, which is closer in similarity to the training data in contrast to the high-burnup datasets. Prediction biases generally do not show a strong directional preference except for energy groups with very low flux values.

An important feature of the CNN trained in this work is the relatively small number of assemblies in its training dataset. CNNs in previous studies were trained using 4400 [9] and 5568 [10] unique assemblies, while the CNN in this work was trained with 300 assemblies. Despite this, the CNN achieved similar accuracy as measured by the fraction of predictions below one and two Monte Carlo standard deviations, as shown in Figure 8, Figure 10, Figure 12, and Figure 14. The neutron flux tallies in this work, however, were recorded on a mesh with spatial elements less than a third of the width of the meshes in [9,10]. By pixel count, the training data in this work would be comparable.

4. Conclusions and Future Work

A residual convolutional neural network was trained on neutron flux tally data computed using OpenMC. The training data consisted of 250 samples of low- and high-resolution flux tallies from fast reactor simulations and 50 samples from thermal spectrum simulations. All training data samples consisted of hexagonal assemblies with two to five rings of pins, with fuel burnup up to 180 MWd/kgIHM. Validation data consisted of eight samples of fast reactor assemblies with 10 or 11 rings of pins and fuel burnup up to 180 MWd/kgIHM. All samples in the training and validation data were heterogeneous procedurally generated assemblies using Latin hypercube sampling to select assembly- and pin-level geometric and material parameters. To test the CNN, eight procedurally generated heterogeneous fast reactor assemblies were simulated with burnup levels from 220 to 400 MWd/kgIHM and 10 or 11 rings of pins. The burnup range was selected to test CNN’s ability to generalize in samples with characteristics that were well outside those encountered in the training data. The CNN was also tested on a simulation of a realistic ESFR assembly with fresh mixed-oxide fuel.

The results show that across all test and validation data, the fraction of predictions outside one (two) standard deviation was 36% (8.6%) and 31% (6.8%), respectively. The residuals and fraction of predictions outside one and two standard deviations were also given by the energy group for individual test datasets. In all test data and energy groups, the residuals were smaller than or close to the Monte Carlo uncertainty on an absolute basis. On a relative basis, this is also the case, with the exception being at energies above 20 MeV, where the neutron flux was close to zero.

Although the CNN trained in this work can make accurate predictions on test data that are considerably dissimilar to the training data, it is limited to specific resolutions. The CNN can upsample 1 × 1 mm tallies to 0.5 × 0.5 mm accurately, but a CNN that can operate on a wider range of input and output resolutions would be of significant utility. Similarly, a CNN with the ability to upsample between different input and output energy resolutions would also be valuable. Similar neural network models, such as generative adversarial networks (GANs), could be explored to enable this capability. GANs have been shown previously to be effective in the super-resolution of climatological data [17], and similar benefits of GANs could be realized in the context of neutron transport simulations. Incorporating uncertainty quantification methods to enable the neural network to estimate the confidence in upsampled predictions would also be essential before its use in neutron Monte Carlo applications. Ensemble methods have been shown to provide uncertainty quantification in simulations of neutron flux monitors [18] and could be adapted for upsampling in a similar way. Finally, the potential for the flexible and accurate upsampling of neutron flux tallies using additional input data that would be available a priori has not been explored. Another CNN input layer could accept a spatial map of the materials used as input to the Monte Carlo simulation. This input layer would be connected to the existing input and encode valuable additional information that would be propagated to downstream layers. Alternatively, or in addition, other types of prior information, such as uncertainties and few-group cross-sections, could be encoded in CNN input layers.