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Neural Bayesian Online Calibration with Deficient Simulators for MADMAX Dark Matter Experiment

This paper introduces a Neural Bayesian online calibration framework that leverages pre-calibration data and normalizing flows to optimize hardware settings and infer latent scientific quantities for experiments using deficient simulators, demonstrating its application to calibrate the MADMAX dark matter experiment's disk-booster simulator and infer the axion-signal power boost factor.

Original authors: Nabil Salama

Published 2026-10-20

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Nabil Salama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Dark matter is the invisible scaffolding of the universe, a substance that does not emit light but reveals its presence through the gravity it exerts on stars and galaxies. While its existence is well-established, the microscopic nature of this substance remains one of physics' greatest mysteries. One leading theory suggests that dark matter consists of axions, hypothetical particles that are incredibly light and interact very weakly with ordinary matter. To find them, scientists build sensitive detectors designed to catch these elusive particles as they convert into electromagnetic waves under strong magnetic fields. However, these experiments are incredibly difficult to tune. The equipment must be positioned with microscopic precision, and the theoretical models used to predict how the machine should behave often fall short because they cannot account for every real-world imperfection, such as tiny manufacturing flaws or complex electromagnetic effects. When the computer model does not perfectly match the physical machine, finding the best settings becomes a guessing game that risks missing the signal entirely.

A researcher at the University of Hamburg has developed a new way to solve this problem, specifically for the MAgnetized Disk and Mirror Axion eXperiment, known as MADMAX. This experiment uses a stack of dielectric disks to amplify the faint signal from axions, but the gap between the disks must be set with extreme accuracy to ensure the waves interfere constructively. The challenge is that the most important measure of success—the power boost factor—cannot be measured directly. Instead, scientists can only measure the reflectivity of the setup, a related but different quantity. Furthermore, the computer simulations used to design the disk spacing are "deficient," meaning they omit physical effects that become significant at the precision required by the experiment. Relying on these imperfect models alone would lead to suboptimal configurations and potentially blind the experiment to the very signal it seeks.

To bridge this gap, the researcher introduced a framework that combines physical measurements with machine learning to calibrate the experiment in real time. The process begins with a pre-calibration phase where the hardware is moved to various positions, and the team records the difference between where the machine was commanded to go and where it actually ended up, as well as the resulting measurements. This data is used to train a neural network, a type of artificial intelligence, to learn a "corrected" version of the simulator. This network learns to predict the effective parameters of the machine that would produce the observed results, effectively teaching the computer how the real world deviates from the idealized model. The system uses a technique called a normalizing flow, which allows it to quickly estimate the probability of different physical states based on a history of measurements, rather than recalculating everything from scratch each time.

Once the machine is trained, it enters an iterative calibration run. The system starts with an initial configuration and begins to probe the experimental setup. For each step, it looks at the entire history of past measurements to update its understanding of the machine's true state. It then decides where to move next by balancing two goals: improving the current signal strength and gaining new information about the system. This decision-making process is guided by a mathematical strategy that weighs the expected improvement against the value of reducing uncertainty. The system accounts for the fact that the hardware might not land exactly where commanded, simulating thousands of possible outcomes to choose the next move that offers the best chance of finding the optimal configuration. This approach allows the experiment to navigate the complex landscape of physical settings without needing a perfect theoretical model to start with.

The researcher applied this method to the MADMAX experiment, using the measurable reflectivity of the setup to calibrate the simulator and infer the unobservable power boost factor. The results show that the framework successfully learns the relationship between the commanded settings and the physical reality, even when the underlying simulator is flawed. By continuously updating its internal model based on real-world data, the system can identify the hardware configuration that maximizes the signal enhancement. While the experimental validation on the actual MADMAX apparatus is still ongoing, the method provides a robust pathway for optimizing complex scientific instruments where traditional modeling fails. This work demonstrates that by letting the machine learn from its own mistakes and measurements, scientists can overcome the limitations of imperfect simulations to push the boundaries of discovery in the search for dark matter.

Technical Summary: Neural Bayesian Online Calibration with Deficient Simulators for MADMAX Dark Matter Experiment

Problem Statement

Scientific experiments frequently rely on "deficient simulators"—models that omit physical effects relevant at the required precision, leading to systematic deviations between predictions and real apparatus behavior. In the context of the MAgnetized Disk and Mirror Axion eXperiment (MADMAX), the goal is to optimize the configuration of dielectric disks to maximize the power boost factor ( $\beta^2$ ) for axion detection. However, $\beta^2$ is a latent (unobservable) quantity, while only the reflectivity ( $\Gamma$ ) is directly measurable.

Conventional optimization or parameter fitting approaches fail in this context for two primary reasons:

Model Discrepancy: The best-fitting simulator parameters may not absorb model errors (e.g., finite-size effects, diffraction, antenna reflections) in a physically meaningful or identifiable way.
Hardware Uncertainty: Experimental configurations derived from calculations cannot be reproduced in the laboratory with perfect precision due to mechanical positioning limits and repeatability issues.

The paper addresses the challenge of performing online calibration and inference when only a deficient simulator is available, at least one quantity is directly measurable, and the quantity of interest is latent.

Methodology

The proposed framework is a three-stage Bayesian online calibration procedure that integrates physical measurements with machine learning to learn a "calibrated effective simulator."

1. Pre-Calibration Data Campaign

Before the main calibration, a data campaign is conducted to characterize the system:

Landing Model: By repeatedly driving hardware to commanded positions $c$ and measuring the realized physical positions $x$ , a landing model $p_L(x|c)$ is established. This accounts for the fact that the realized query location is random even when the command is known.
Measurement Model: Repeated measurements at fixed positions allow for the estimation of measurement uncertainty $\Sigma_{\hat{U}_0}$ .
Data Collection: A dataset $D_{cal}$ is recorded consisting of commanded positions, realized physical positions, and measured reflectivity values.

2. Pre-Calibration Training

The core of the method involves learning an effective simulator parameterization $\theta(x)$ that describes the true system state, rather than just physical parameters.

Effective Prior Learning: A conditional prior $p_\psi(\theta|x)$ is learned from the pre-calibration data. This is achieved by fitting a model (e.g., a Gaussian mixture) to maximize the likelihood of observed measurements given the simulator output and the prior.
Amortized Posterior Inference: A conditional normalizing flow $F_\phi$ $F_{ϕ}$ is trained to act as an amortized posterior estimator.
- Training Data: Synthetic measurement histories are generated by sampling from the learned prior, the deficient simulator, and the measurement likelihood.
- Architecture: The network consists of an encoder that transforms variable-length measurement histories into a context vector (using Deep-Sets style aggregation for permutation invariance) and a normalizing flow that proposes parameter sets from the posterior distribution.
- Objective: The network minimizes the negative log-likelihood of the true parameters given the synthetic history.

3. Iterative Calibration Run

The online calibration loop uses the learned models to iteratively select hardware commands to optimize the latent quantity $U_i$ (e.g., power boost factor).

Posterior Estimation: At each step, the normalizing flow generates particles (parameter samples) conditioned on the current measurement history. These are re-weighted using the learned conditional prior and the measurement likelihood to approximate the true posterior.
Acquisition Strategy: The next command $c_{t+1}$ $c_{t + 1}$ is selected by balancing two criteria:
1. Expected Improvement (EI): Estimates the potential gain in the objective function $J(U_i)$ relative to the current best.
2. Information Gain (IG): Estimates the reduction in entropy regarding which command is optimal, favoring measurements that resolve uncertainty about the global optimizer.
Balancing: A time-varying coefficient $\alpha_t$ controls the trade-off between exploitation (improving the current value) and exploration (reducing uncertainty), shifting from improvement-focused to information-focused selection over time.
Constraint: Commands are only considered if their landing distribution remains within the region covered by the learned prior.

Key Contributions

Integration of Amortized Inference and Bayesian Optimization: The paper integrates conditional density estimation (via normalizing flows) with sequential acquisition criteria into a unified online hardware-calibration workflow.
Handling Deficient Simulators: The method explicitly addresses model discrepancy by learning "effective" simulator parameters that describe the true system state, rather than attempting to fit physical parameters that may be unidentifiable due to model errors.
Propagation of Uncertainties: The framework rigorously propagates both measurement uncertainties and hardware positioning uncertainties (via the landing model) through the optimization process.
Latent Quantity Inference: It enables the inference of unobservable scientific quantities (like the axion signal boost) by leveraging a directly measurable proxy (reflectivity) within a shared parameter space.

Results and Validation

Application to MADMAX: The framework is applied to the MADMAX experiment to calibrate the disk-booster simulator and infer the power boost factor $\beta^2$ .
Experimental Status: The paper states that experimental validation on the actual MADMAX setup is ongoing. Consequently, no final numerical results or performance benchmarks against the physical experiment are presented in this preprint.
Synthetic Validation: The training data for the neural posterior estimator is generated synthetically, and the paper notes that the adequacy of these synthetic histories must be validated against held-out real measurements to ensure the estimator does not become biased or overconfident due to simulator misspecification.

Significance and Claims

The paper claims to provide a robust procedure for optimizing experimental configurations and inferring quantities of interest in scenarios where conventional approaches fail due to model error and hardware limitations. By combining Bayesian calibration with deep learning, the method allows for the efficient use of scarce experimental data to learn effective simulator parameters.

The author emphasizes that this approach is particularly significant for experiments like MADMAX, where the quantity of interest is hidden and the simulator is inherently deficient. The method aims to align hardware settings to optimize the latent quantity while providing uncertainty-aware estimates, thereby meeting scientific requirements that purely simulation-based or standard fitting approaches cannot achieve. The work is presented as a preprint with experimental validation in progress, suggesting that the full empirical verification of the framework's efficacy is a future step.