GokuNEmu: 10D Neural-Network Emulator

• GokuNEmu is a ten-dimensional neural-network emulator that replaces computationally expensive N-body simulations for predicting the nonlinear matter power spectrum P(k,z).
• It employs a two-step multifidelity approach, combining low- and high-fidelity data with neural-network corrections to achieve ~0.5% mean error across broad cosmological parameter spaces.
• Its rapid evaluation (~2 ms per cosmology) makes it ideal for MCMC, nested sampling, and joint analyses in surveys like DESI, LSST, and Euclid.

Searching arXiv for GokuNEmu and closely related cosmology papers to ground the article. GokuNEmu is a ten-dimensional neural-network emulator for the nonlinear matter power spectrum $P(k,z)$, built on the Goku $N$-body simulation suite and the T2N-MusE multifidelity emulation framework to support next-generation cosmological inference. It predicts $P(k,z)$ over $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$ and $0 \le z \le 3$ with $\sim 0.5\%$ average accuracy, while evaluating a single cosmology in $\sim 2$ milliseconds on a laptop. Its parameter domain extends beyond flat $\Lambda$CDM to include dynamical dark energy, massive neutrinos, the effective number of neutrinos, and running of the scalar spectral index, with prior coverage explicitly described as broad enough to test recent DESI dynamical dark energy constraints (Yang et al., 9 Jul 2025).

1. Scientific target and formal definition

GokuNEmu addresses a standard bottleneck in precision large-scale-structure analysis: surveys such as DESI, LSST, Euclid, the Roman Space Telescope, and CSST require repeated, high-precision evaluations of the nonlinear matter power spectrum across extended cosmological parameter spaces, whereas a new high-resolution $N$-body simulation at each likelihood point is computationally infeasible. In the GokuNEmu formulation, the emulator replaces those simulations with a trained surrogate that operates inside the prior box and returns $P(k,z)$ in $N$0 seconds rather than thousands of CPU hours (Yang et al., 9 Jul 2025).

The target observable is the fully nonlinear matter power spectrum measured directly from $N$1-body simulations rather than a linear-theory spectrum. The underlying matter density contrast is

$N$2

with power spectrum defined by

$N$3

GokuNEmu therefore belongs to the class of simulation-based nonlinear emulators rather than fitting-function models such as Halofit or HMcode (Yang et al., 9 Jul 2025).

A central distinction from earlier cosmological emulators is dimensionality coupled to broad prior coverage. The emulator spans a 10D space with explicit dark-energy and neutrino-sector extensions, and its evaluation speed is substantially faster than earlier Gaussian-process-based approaches, including its predecessor GokuEmu (Yang et al., 9 Jul 2025).

2. Cosmological parameterization and validity domain

GokuNEmu assumes a spatially flat universe and emulates $N$4 as a function of ten cosmological parameters. The parameterization combines five flat $N$5CDM-like parameters with five extensions.

Group	Parameters
Base flat $N$6CDM-like	$N$7
Extensions	$N$8

The dark-energy sector uses the CPL form

$N$9

while the running parameter is defined as $P(k,z)$0. The wide simulation box Goku-W covers $P(k,z)$1 to $P(k,z)$2, $P(k,z)$3 to $P(k,z)$4, $P(k,z)$5 to $P(k,z)$6, $P(k,z)$7 to $P(k,z)$8, $P(k,z)$9 to $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$0, $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$1 to $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$2, $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$3 to $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$4, $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$5 to $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$6, $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$7 to $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$8, and $0.006 \le k/(h\,\mathrm{Mpc}^{-1}) \le 10$9 to $0 \le z \le 3$0. The nested narrow box Goku-N concentrates on a higher-likelihood region with correspondingly tighter intervals (Yang et al., 9 Jul 2025).

The extensions are motivated by distinct physical imprints on $0 \le z \le 3$1. Dynamical dark energy alters the expansion history and structure-growth rate; massive neutrinos suppress small-scale power through free-streaming; $0 \le z \le 3$2 modifies the early expansion rate and thus the transfer function; and $0 \le z \le 3$3 changes the detailed scale dependence inherited from primordial fluctuations. The broad prior coverage is specifically described as safely enclosing the full $0 \le z \le 3$4 regions from DESI and DES dynamical dark-energy analyses in the $0 \le z \le 3$5–$0 \le z \le 3$6 plane, while the $0 \le z \le 3$7 prior up to $0 \le z \le 3$8 is stated to cover cosmological limits and the KATRIN bound $0 \le z \le 3$9 eV at 90% C.L. (Yang et al., 9 Jul 2025).

The modeled regime extends over $\sim 0.5\%$0 and $\sim 0.5\%$1. The low-$\sim 0.5\%$2 end is mostly linear or weakly nonlinear, $\sim 0.5\%$3–$\sim 0.5\%$4 is quasi-linear to nonlinear, and $\sim 0.5\%$5 reaches the deeply nonlinear regime, which is particularly relevant for galaxy clustering and weak lensing in upcoming surveys (Yang et al., 9 Jul 2025).

3. Simulation basis and multifidelity data model

The emulator is trained on the Goku simulation suite, a 10-dimensional cosmological $\sim 0.5\%$6-body design constructed with MP-Gadget and organized through nested wide and narrow hypercubes sampled by a Sliced Latin Hypercube Design. The suite combines a small set of expensive high-fidelity simulations with a much larger set of cheaper low-fidelity runs, an arrangement originally developed in the MF-Box framework and publicized with the Gaussian-process predecessor GokuEmu (Yang et al., 10 Jan 2025).

The high-fidelity component consists of 36 simulations in total: 21 in Goku-W and 15 in Goku-N. Each has box size $\sim 0.5\%$7 and $\sim 0.5\%$8 dark-matter particles. The low-fidelity component comprises 1128 paired simulations. The L1 node uses the same box size as HF, $\sim 0.5\%$9, but only $\sim 2$0 particles, so it captures large scales economically but loses high-$\sim 2$1 convergence. The L2 node uses $\sim 2$2 with $\sim 2$3 particles, providing better effective resolution at high $\sim 2$4 while missing the largest modes. The low-fidelity runs densely sample parameter space and the high-fidelity simulations supply the correction needed for accurate emulation (Yang et al., 9 Jul 2025).

The original Goku simulation paper further specifies that the simulations use MP-Gadget, begin from CLASS-based initial conditions, and implement massive neutrinos through the linear response method of Ali-Haimoud & Bird rather than neutrino particles. In that setup, neutrinos contribute to the long-range gravitational potential while short-range neutrino forces are neglected because free-streaming suppresses them below the PM-cell scale (Yang et al., 10 Jan 2025).

Power spectra are measured from each simulation at six original snapshots, $\sim 2$5, plus more than one hundred intermediate outputs that are interpolated down to 28 additional redshift bins, yielding 34 emulated redshift bins in total. The authors report that linear interpolation in $\sim 2$6 between raw outputs reproduces the simulation spectra to better than $\sim 2$7 in $\sim 2$8 (Yang et al., 9 Jul 2025).

Because L1 and L2 converge in different $\sim 2$9-ranges, GokuNEmu is assembled from two component emulators. Emu1 uses L1 and HF data for $\Lambda$0, while Emu2 uses L2 and HF data for $\Lambda$1. The final prediction is blended in the overlap region through

$\Lambda$2

with $\Lambda$3 a sigmoid weight function. A further pairing-and-fixing correction, following Angulo and Pontzen, is applied as a universal correction to reduce residual cosmic variance (Yang et al., 9 Jul 2025).

4. Neural-network construction and T2N-MusE methodology

GokuNEmu is implemented within T2N-MusE, a multifidelity framework based on fully connected neural networks. At the conceptual level, the emulator takes as input the ten cosmological parameters together with redshift $\Lambda$4 and returns the nonlinear power spectrum values on a fixed $\Lambda$5-grid. Its core design is a two-step multifidelity architecture in which one network learns the low-fidelity spectrum and a second network learns the high-fidelity correction as a ratio (Yang et al., 9 Jul 2025).

The two-step mapping is

$\Lambda$6

This preserves the multifidelity logic of earlier Gaussian-process constructions while replacing GP regression with neural networks. The short GokuNEmu paper emphasizes the ratio-based correction and the use of separate low-$\Lambda$7 and high-$\Lambda$8 emulators, whereas the dedicated T2N-MusE methodology paper characterizes the framework more generally by four ingredients: a 2-step multifidelity architecture, a 2-stage Bayesian hyperparameter optimization, a 2-phase $\Lambda$9-fold training strategy, and a per-$N$0 principal component analysis strategy (Yang et al., 9 Jul 2025, Yang et al., 9 Jul 2025).

Hyperparameter selection is carried out by two-stage Bayesian optimization over the number of layers, the number of neurons per layer, and the $N$1 regularization strength. Training of the low-fidelity network uses a two-phase procedure: multiple random seeds are first explored to identify favorable minima, and $N$2-fold cross-validation is then run from the best initial weights to improve generalization and reduce overfitting. The T2N-MusE paper reports that these architectural and optimization choices reduce validation error by more than a factor of five relative to previous work on the same Goku data, and that the resulting production emulator is GokuNEmu (Yang et al., 9 Jul 2025).

The networks are implemented in PyTorch and use standard modern FCNN components, including the Adam optimizer, weight decay expressed as decoupled $N$3 regularization, and advanced activation functions in the Swish/SiLU family. The short letter does not state the exact layer counts or neuron widths, and it does not dwell on preprocessing beyond the multifidelity construction itself (Yang et al., 9 Jul 2025).

5. Validation, inference speed, and comparison with other emulators

Accuracy is quantified through a relative mean absolute error between emulator predictions and held-out high-fidelity simulation outputs, averaged over $N$4 and $N$5. Validation uses leave-one-out cross-validation on the HF set: for each held-out HF cosmology, the corresponding simulation is removed, the multifidelity networks are retrained on the remaining HF and LF data, and predictions are compared with the withheld HF spectrum (Yang et al., 9 Jul 2025).

On the wide box Goku-W, the mean error over all test HF cosmologies, 34 redshifts, and 137 $N$6-bins is reported as $N$7. On the narrow box Goku-N, the mean error is $N$8. Error is typically below $N$9 for almost all $P(k,z)$0 and $P(k,z)$1, with the largest deviations appearing at the lowest redshift and highest $P(k,z)$2, where nonlinearity and resolution demands are most severe. At $P(k,z)$3 and $P(k,z)$4, some cosmologies rise slightly above $P(k,z)$5 but remain at the percent level (Yang et al., 9 Jul 2025).

These results represent a substantial improvement over the Gaussian-process predecessor GokuEmu. For the same wide-box training data, GokuEmu had mean error $P(k,z)$6, whereas GokuNEmu reaches $P(k,z)$7. In the narrow box, GokuEmu was already very accurate, but GokuNEmu is reported to reduce errors at high $P(k,z)$8 and low $P(k,z)$9 further (Yang et al., 9 Jul 2025). The original GokuEmu paper had described the earlier emulator as achieving percent-level accuracy in high-likelihood regions and $N$00 accuracy across broader parameter ranges, while reducing simulation cost by 94% relative to single-fidelity strategies (Yang et al., 10 Jan 2025).

Inference latency is another defining feature. GokuNEmu requires $N$01 ms per cosmology on a standard laptop CPU for the full $N$02 at one redshift. The same source compares this with $N$03 ms for EuclidEmulator2, $N$04 ms for the CSST emulator, “a few seconds” for GP-based GokuEmu, and $N$05–$N$06 CPU-seconds for a full high-fidelity $N$07-body simulation (Yang et al., 9 Jul 2025).

Relative to competing tools, the defining combination is dimensionality, prior width, speed, and low error. The emulator includes $N$08, $N$09, $N$10, $N$11, and $N$12 simultaneously; its $N$13–$N$14 box is described as safely enclosing the full $N$15 DESI and DES regions; and its neutrino prior extends to $N$16, wider than EuclidEmulator2. On that basis, the GokuNEmu paper describes it as the only matter power spectrum emulator capable of testing recent dynamical dark energy constraints from DESI (Yang et al., 9 Jul 2025).

6. Applications, limitations, and terminological context

The intended use cases are fast forward modeling in MCMC, nested sampling, Fisher forecasting, and broader sensitivity analysis for joint cosmological inference. Because the emulator returns nonlinear $N$17 across a 10D extended parameter space, it is suited to joint fits involving combinations such as CMB, BAO, supernovae, and large-scale structure, as well as survey-specific analyses for LSST, Euclid, Roman, and CSST (Yang et al., 9 Jul 2025).

A parameter-sensitivity study in the GokuNEmu paper underscores why the nonlinear regime matters for extension models. Using a reference cosmology

$N$18

and a DESI-like cosmology with $N$19, the study finds that at $N$20 the most sensitive scale to changes in $N$21, $N$22, and $N$23 lies near $N$24, i.e. deep in the nonlinear regime. It also shows that changes in $N$25 and $N$26 can mimic lowering $N$27 below the oscillation-experiment minimum, suggesting that fixing $N$28 and $N$29 can artificially push inferred neutrino masses downward (Yang et al., 9 Jul 2025).

The validity domain is explicitly bounded. The emulator is trained strictly within the Goku-W and Goku-N parameter boxes and extrapolation beyond them is not recommended. It is validated only for $N$30. The underlying simulations are gravity-only dark-matter $N$31-body runs with no baryonic feedback, so real-data analyses at $N$32–$N$33 require an additional treatment of baryons, such as baryonification or baryonic correction schemes. Dark-energy clustering and modified-gravity effects beyond the CPL $N$34–$N$35 model are not included, and nonstandard models such as warm dark matter or early dark energy are outside scope (Yang et al., 9 Jul 2025).

The software is publicly available in PyTorch at the repository listed by the authors, and the intended workflow is straightforward: specify the cosmological parameter vector, provide a redshift, obtain $N$36 on the internal $N$37-grid, and interpolate in $N$38 if needed (Yang et al., 9 Jul 2025).

A recurring source of confusion is nomenclature. Within cosmological emulation, GokuNEmu denotes the 2025 neural-network emulator for the nonlinear matter power spectrum, whereas GokuEmu denotes the earlier Gaussian-process emulator built on the same Goku simulation suite (Yang et al., 10 Jan 2025, Yang et al., 9 Jul 2025). The name should also not be conflated with unrelated arXiv projects titled “Goku” in flow-based image/video generation or instruction-based video editing (Chen et al., 7 Feb 2025, Liang et al., 29 Jun 2026).