nuMax: Accelerator, Simulator & Optimization

• nuMax is a multidisciplinary technical system encompassing a neutrino factory accelerator, a GPU-parallel reinforcement learning simulator, and a distributed network utility maximization algorithm.
• The NuMAX Neutrino Factory employs advanced SRF acceleration, dual-use linacs, and ionization cooling to generate high-intensity, precisely characterized neutrino beams for frontier physics research.
• The simulator and optimization variants leverage GPU acceleration and distributed Newton-GaBP methods to achieve significant speedups and robust convergence in scalable reinforcement learning and resource allocation.

nuMax denotes several distinct technical systems across high-energy physics, large-scale optimization, and machine learning for simulation. This article covers the principal implementations of nuMax as found in published literature: (1) the NuMAX Neutrino Factory, a staged muon accelerator complex and flagship proposal for next-generation neutrino physics; (2) nuMax, a GPU-parallel simulation backend for scalable reinforcement learning in diffusion-based planners; and (3) nuMax as a numerically robust, distributed algorithm for large-scale network utility maximization (NUM).

1. NuMAX (Neutrinos from Muon Accelerator compleX): Facility Concept and Physics Scope

NuMAX is a staged accelerator facility proposal designed for precision neutrino studies, muon-based intensity-frontier measurements, and as a development path toward high-energy muon colliders. The initial configuration is sited at Fermilab and leverages synergies with PIP-II proton driver infrastructure and the DUNE far detector (Delahaye et al., 2018). The staged program consists of:

• Stage 0 ("nuSTORM"): Short-baseline ring for muon storage at ≈3.8 GeV, targeting sterile-neutrino and cross-section physics.
• Stage 1 ("NuMAX Commissioning"): 5 GeV Neutrino Factory with 1 MW proton driver, no cooling, 10 kt magnetized liquid-argon detector at 1300 km.
• Stage 2 ("NuMAX Baseline"): Addition of modest six-dimensional muon cooling, maintaining 1 MW driver but ×4 flux gain in usable muons.
• Stage 3 ("NuMAX⁺"): Upgrade to full-cooling channel and higher proton power (2.75 MW), with >5×10²⁰ ν/yr at the far detector and enhanced detector mass.

Key technical concepts underpin both cost and performance: advanced superconducting RF acceleration, dual-use linac modules (serving both proton and muon beams), dogbone recirculating linac architectures, high-field solenoidal capture and ionization cooling, and a flexible ring infrastructure that supports transition from neutrino factory to multi-TeV muon collider operation (Bogacz, 2017).

2. Accelerator Architecture and Technical Innovations

The NuMAX acceleration complex is built around two main schemes after the front-end linac (Bogacz, 2017):

A. Dual-Use 650 MHz Linac:

• The linac operates at 325 MHz up to 1.25 GeV, transitioning to 650 MHz at higher energy as transverse phase-space shrinks.
• The same SRF structure accelerates both H⁻ and μ⁺/μ⁻, reducing capital costs by ~30% relative to a dedicated muon chain.
• Solenoid-focused FOFO cells are employed at low energy, with FODO quadrupole optics above 2.5 GeV.
• Achieves ≳85% muon survival for 1.25→5 GeV, with real estate gradient up to 25 MV/m.

B. Dogbone Recirculating Linac Accelerator (RLA):

• 650 MHz SRF linac is reused for ~4.5 passes, drastically reducing installed linac length.
• Fixed-energy arcs are required for each pass. Recent designs consider non-scaling FFAG-like arcs to reduce cost and enable energy flexibility.
• Typical total acceleration time is ~10 μs, yielding P_surv ≈ 0.85–0.88 through 5 GeV.
• RLA offers natural phase-space rotation and upgrade path for higher energy (multi-TeV), but is ~10–15% more expensive than the dual-use linac in the 5 GeV regime.

Chicane and Matching Optics:

• Double charge-separation chicane and longitudinal phase-space manipulations synchronize μ⁺/μ⁻ RF phases and match large-emittance front-end beams to high-frequency, small-bucket subsequent linacs.

Cooling/Acceptance Trade-off:

• Large normalized transverse (20 mm·rad) and longitudinal acceptances relax cooling demands but dictate large-aperture, low-frequency initial acceleration stages.
• Moderate cooling is retained to minimize cost and complexity while preserving muon yield.

3. Physics Output and Analytical Formalism

NuMAX produces precisely characterized neutrino beams from stored muon decays, enabling leading sensitivity in CP-violation, mass ordering, and non-standard interactions at long baselines (Delahaye et al., 2018).

Neutrino Flux

The differential neutrino flux at distance L is given by $$\Phi_\nu(E, \Omega) = \frac{N_\mu\,f}{4\pi\,L^2} \frac{d^2N_\nu}{dE d\Omega}$$ where $N_\mu$ is the total stored muons, and $f$ is the straight-section/circumference fraction (≈0.35).

Event Rates

The expected number of events, integrating over the flux and cross-section for a detector of fiducial mass $M_{det}$, is $$N_{ev} = N_{tgt} \int dE\, \Phi_{\nu_\alpha}(E)\, \sigma_{\nu_\alpha\to\beta}(E)\, P_{\alpha\beta}(E)\, \varepsilon(E)$$ where $N_{tgt} = \frac{M_{det}\,N_A}{A}$ and $P_{\alpha\beta}(E)$ encodes oscillation physics.

Performance

NuMAX⁺ with 5×10²⁰ ν/yr, 30 kt detector, and baseline of 1300 km achieves δ_CP phase precision better than 5°, substantially surpassing conventional superbeams.

Staged upgrades support strategic transitions to a Higgs Factory or multi-TeV muon collider using the same backbone of SRF, cooling, and recirculating optics, ensuring flexibility for future physics directions.

4. nuMax: GPU-parallel Simulation for Reinforcement Fine-Tuning

In a distinct research context, nuMax is a JAX-native, GPU-parallel traffic simulator purpose-built for large-scale closed-loop reinforcement fine-tuning of diffusion-based planners in autonomous driving (Li et al., 19 Jan 2026). Key architectural elements include:

• Scenario Pre-Caching:

Conversion of nuPlan/Waymax HD-maps and scenario archives into TFRecords eliminates I/O bottlenecks by enabling rapid in-memory slicing rather than heavy per-step SQL/GeoPandas queries.

• Batched Controllers and Reward:

JAX-compiled, vectorized LQR-Bicycle controllers and fused reward kernels facilitate parallel rollouts across B environments and G action hypotheses per step.

• Hybrid PyTorch–JAX DDP:

Observations transfer from a single host JAX environment to PyTorch DDP workers, which run denoising and policy updates. The system ensures minimal host-device synchronization overhead and avoids XLA static-shape conflicts.

Performance Benchmark:

Simulator	Hardware	Latency (ms/step)	Env-Steps/s (B=128×32)	Speedup
nuPlan	16-core Xeon	0.98	4.1×10³	1×
nuMax	NVIDIA A100	0.10	41×10³	9.8×

Wall-clock time for collecting 1 M environment steps is reduced from >5 h (CPU nuPlan) to <30 min (GPU nuMax).

Limitations:

nuMax currently relies on static-shaped tensors due to XLA, with future work focused on dynamic scenario caching, distributed multi-device support, and full reactive background agent simulation.

5. nuMax in Distributed Network Optimization

An additional, mathematically distinct instantiation of "nuMax" appears as a numerically robust distributed algorithm for network utility maximization (NUM) (0901.2684). The problem is formulated as $$\begin{aligned} & \underset{f\in\mathbb R^n}{\text{maximize}} && \sum_{j=1}^n U_j(f_j) \ & \text{subject to} && Rf \leq c,\, f \geq 0 \end{aligned}$$ where $U_j$ are concave utilities and $R$ is the routing matrix.

Algorithmic Approach:

The core of nuMax is an interior-point Newton solver, where:

• The Newton system is solved via Gaussian belief propagation (GaBP), exploiting the sparsity and locality of network graphs and enabling fully distributed operation.
• Each computational node maintains and updates local precision and mean messages, with only neighboring communication required.
• Empirical results demonstrate that truncated Newton + GaBP achieves 10–20× fewer inner iterations and much improved robustness relative to preconditioned conjugate gradient (PCG), as well as orders-of-magnitude faster convergence than classical dual decomposition.

Scalability:

On problems up to $n=10^5$ flows, $m=2\times10^5$ links, nuMax attains duality gaps <10⁻⁴ in ≈11 Newton steps, with 7–9 GaBP iterations per step—enabling real-time distributed computation at scale.

6. Distinctions and Lexical Notes

"nuMax" refers to multiple technical entities that are unrelated in implementation and application domain:

• NuMAX (Neutrino Factory): Accelerator complex, muon/neutrino physics (Delahaye et al., 2018, Bogacz, 2017).
• nuMax (simulator): High-throughput simulator for RL in AV planning (Li et al., 19 Jan 2026).
• nuMax (network optimization): Distributed interior-point solver for large-scale NUM (0901.2684).

Context should be inferred from disciplinary cues: NuMAX (all-caps) is universally used for the Fermilab-related facility, while lower-case nuMax is reserved for software systems in machine learning or optimization.

7. Future Prospects

The NuMAX Neutrino Factory and its accelerator technology underpin ongoing R&D toward ultimate muon-based energy-frontier facilities in both neutrino and collider physics scenarios (Delahaye et al., 2018, Bogacz, 2017). GPU-parallel simulators such as nuMax enable new regimes of scalability and efficiency in simulation-driven reinforcement learning (Li et al., 19 Jan 2026), which is of growing importance in robotics, AV planning, and RL-in-the-loop research. Distributed optimization architectures exemplified by nuMax's Num-based Newton-GaBP approach remain central to large-scale resource allocation, offering robustness and speed that match or surpass best-known centralized solvers (0901.2684).