NPLM: New Physics Learning Machine

• New Physics Learning Machine (NPLM) is a machine-learning goodness-of-fit method that implements Neyman–Pearson likelihood-ratio tests for model-independent anomaly detection in collider experiments.
• It leverages both neural networks and kernel methods to learn flexible density deformations, enhancing sensitivity to localized deviations in high-dimensional data.
• The approach incorporates rigorous calibration with toy datasets, aggregated testing, and multi-hyperparameter strategies to robustly identify discrepancies in background-dominated scenarios.

The New Physics Learning Machine (NPLM) is a machine-learning-based goodness-of-fit methodology that implements the Neyman–Pearson likelihood-ratio strategy in a signal-agnostic setting, primarily for collider searches in high-energy physics. It treats goodness of fit as an asymmetric two-sample problem: the observed dataset is compared with a large reference sample drawn from an expected distribution, and the alternative hypothesis is not fixed analytically but learned from data as a flexible deformation of the reference density. In this form, NPLM is simultaneously a hypothesis test, a density-ratio estimator, and a diagnostic for locating discrepant regions of phase space (D'Agnolo et al., 2018, Grosso et al., 2023).

1. Origins and problem setting

NPLM was introduced for model-independent new physics searches in situations where one has a trusted reference model, typically the Standard Model or a background-only prediction, and wishes to test whether observed data are compatible with it without assuming a specific signal model in advance. The original proposal used neural networks as flexible function approximators to learn departures from a reference distribution, and later work reformulated the method explicitly as a goodness-of-fit procedure based on Neyman–Pearson testing and extended it to scalable kernel implementations (D'Agnolo et al., 2018, Letizia et al., 2022).

The central problem is background-dominated anomaly detection. New physical effects, if present, are expected to be similar to the reference model in most of phase space, while departures may be localized in small regions, spread as small distortions over large regions, or combine shape and rate effects. NPLM addresses this by asking only whether the data and the reference look like they come from the same distribution. In this sense it is model-independent in the physics sense: the alternative is not derived from a specific theory, but chosen for flexibility and technical convenience (D'Agnolo, 2018, Grosso et al., 2024).

The method was developed for agnostic searches for new physics at collider experiments, including Beyond-the-Standard-Model searches in LHC data, dimuon benchmarks, and detector-performance monitoring. Later work broadened the scope to quasi-online anomaly-aware summary statistics, resonant anomaly detection, and generative-model validation, while retaining the same statistical core (Grosso et al., 2023, Grosso, 2024, Cappelli et al., 12 Nov 2025).

2. Neyman–Pearson formulation and test statistic

The defining step in NPLM is to write the alternative hypothesis as a deformation of the reference density,

$n(x\mid H_1)=e^{f(x)}\,n(x\mid R),$

where $f(x)$ belongs to a rich parametrized function class and is trained from data. Here $n(x\mid H)=N(H)p(x\mid H)$ denotes the event density normalized to the expected number of events. This parameterization guarantees positivity and embeds the null hypothesis as the special case $f=0$ (Grosso et al., 2023, D'Agnolo et al., 2021).

The statistical construction uses the extended likelihood

$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$

which accounts for both shape information and Poisson fluctuations in the total event count. Maximizing the likelihood under the learned alternative yields a likelihood-ratio-style test statistic. In Monte Carlo form, one widely used expression is

$$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$

and large values indicate that the learned alternative describes the data better than the reference (Grosso et al., 2023).

A closely related formulation writes the observed statistic as

$$t_{\hat w}(S_1) =-2\left[\frac{N(0)}{N_0}\sum_{x\in S_0}\left(1-e^{f_{\hat w}(x)}\right)+\sum_{x\in S_1}f_{\hat w}(x)\right],$$

with $S_0$ the reference sample and $S_1$ the data sample. In both forms, the learned function approximates the log-density ratio

$f(x)\approx \log\frac{n(x\mid 1)}{n(x\mid 0)},$

so the test statistic is an empirical implementation of a maximum-likelihood ratio test with a data-driven alternative (Letizia et al., 2022).

The final goodness-of-fit judgment is based on the $f(x)$0-value,

$f(x)$1

estimated empirically from toy datasets generated under the reference hypothesis. This calibration is essential because the same data are used both to learn the alternative and to evaluate the test statistic (Grosso et al., 2023).

3. Learning machinery, density-ratio estimation, and regularization

The original NPLM implementation used a neural network $f(x)$2 as a universal approximator of the log-density ratio. The training loss was chosen so that minimizing it corresponds to maximizing the likelihood under the alternative. Two equivalent losses appear in the literature: a weighted logistic loss,

$f(x)$3

and a maximum-likelihood loss,

$f(x)$4

In the large-sample limit, the minimizer is the desired log-density ratio (Grosso et al., 2023).

A later reformulation replaced neural networks with kernel methods, specifically the scalable Falkon algorithm with Nyström approximation. In that setting,

$f(x)$5

with Gaussian kernel

$f(x)$6

This preserves the original logic of the method while making training dramatically faster and more resource-efficient. Reported average training times per run were DIMUON: Falkon $f(x)$7 s vs NN $f(x)$8 h, SUSY: Falkon $f(x)$9 s vs NN $n(x\mid H)=N(H)p(x\mid H)$0 h, and HIGGS: Falkon $n(x\mid H)=N(H)p(x\mid H)$1 s vs NN $n(x\mid H)=N(H)p(x\mid H)$2 h; the targeted regime was described as roughly $n(x\mid H)=N(H)p(x\mid H)$3 in time and $n(x\mid H)=N(H)p(x\mid H)$4 in memory (Letizia et al., 2022).

Model-independence is enforced through the treatment of hyperparameters. In the kernel implementation, the main hyperparameters are the number of Nyström centers $n(x\mid H)=N(H)p(x\mid H)$5, the kernel width $n(x\mid H)=N(H)p(x\mid H)$6, and the regularization strength $n(x\mid H)=N(H)p(x\mid H)$7. The literature states that $n(x\mid H)=N(H)p(x\mid H)$8 should be large enough for good approximation accuracy, roughly at least of order $n(x\mid H)=N(H)p(x\mid H)$9, $f=0$0 should be as small as possible while keeping training stable, and $f=0$1 is the most influential parameter because it sets the scale of structure the classifier can resolve. Hyperparameters are tuned only on reference data to avoid bias toward a particular anomaly (Letizia et al., 2022, Grosso et al., 2024).

A frequent misconception is that NPLM is merely a classifier-based two-sample test under another name. The comparison studies argue otherwise. NPLM evaluates the learned statistic on the same data used for training rather than relying on a train-test split, and its test statistic is a likelihood-ratio statistic rather than accuracy or AUC. These features are reported to matter operationally: train-test splitting removes useful information and reduces sensitivity, while likelihood-ratio weighting gives more importance to points in regions where the learned model is confident that the data and reference differ. In the benchmarks studied, this makes NPLM more sensitive to small departures of the data from the expected distribution and not biased toward detecting specific types of anomalies (Grosso et al., 2023).

4. Calibration, nuisance profiling, and multiple testing

Because NPLM uses the observed data to learn the alternative, null calibration is performed by retraining on many toy datasets drawn from the reference model. For a fixed hyperparameter setting, the procedure is: train the NPLM classifier on reference and data, compute the observed statistic, estimate the null distribution by toy experiments that retrain from scratch, and convert the result to a $f=0$2-value and then to a significance through

$f=0$3

One finite-sample corrected empirical $f=0$4-value used in the literature is

$f=0$5

This retraining-based calibration controls both the use of the data twice and the implicit look-elsewhere effect associated with the learned function class (Grosso et al., 2024, D'Agnolo, 2018).

Systematic uncertainties can be incorporated by treating the reference prediction as nuisance-parameter dependent. In that formulation, the alternative becomes

$f=0$6

and the profile-likelihood-ratio statistic is decomposed as

$f=0$7

Here $f=0$8 contains the neural-network deformation and nuisance dependence, while $f=0$9 subtracts the nuisance-only response of the reference model. Auxiliary neural networks are trained to learn nuisance-induced density ratios, and the subtraction is used to restore the correct null behavior. The method works well when the reference sample sufficiently covers the relevant phase space, but can produce outliers if nuisance shifts push the data into regions not populated by the reference sample (D'Agnolo et al., 2021).

A distinct robustness problem arises from hyperparameter dependence, especially the kernel width $$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$0. Because different $$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$1 values are optimal for different anomaly morphologies, multiple-testing strategies have been studied in which several NPLM tests are run and then combined. The methods explored include min-$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$2, prod-$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$3, avg-$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$4, and smax-$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$5. The reported conclusion is that combining tests characterised by distinct choices of hyperparameters achieves performances comparable to the best available test while providing a more uniform response to various types of anomalies. Among the tested rules, min-$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$6 is described as the most consistently robust, whereas smax-$$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$7 is argued to be intrinsically biased because the NPLM test statistic tends to increase for smaller $$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$8 (Grosso et al., 2024).

5. Batched, aggregated, and streaming variants

NPLM has also been adapted to large size samples under computational and storage resource constraints. In the batched formulation, the dataset is split into $$\mathcal{L}(S_1,H)=\frac{e^{-N(H)}}{N_1!}\prod_x n(x\mid H),$$9 batches and an independent NPLM model is trained on each batch. The simplest combination is the sum of the per-batch likelihood-ratio statistics, denoted $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$0. A more effective construction instead aggregates the learned log-density ratios themselves under the assumption that the expected rate of new physical processes is time invariant (Grosso, 2024).

The aggregated log-ratio is defined by averaging the learned density deformations across batches and then taking the logarithm,

$$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$1

which yields a global hypothesis shared over the batches. The resulting test statistic, $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$2, preserves the Neyman–Pearson spirit of NPLM while making the method scalable to large streams or large static datasets. The reported empirical finding is that this method outperforms the simple sum of the independent tests run over the batches, and can recover, or even surpass, the sensitivity of the single test run over the full data (Grosso, 2024).

Two constrained variants were proposed for reduced-storage settings. NPLM-ONE uses all batches for training but only one stored batch for the final test. NPLM-SAT stores only the trained models and uses a saturated binned likelihood-ratio test built from the aggregated model evaluated on the reference points. These variants were framed as anomaly-aware summary statistics suitable for quasi-online data streaming scenarios. In the reported experiments, performance of the saturated construction was comparable to the full aggregated test when the signal lies in dense regions of the reference support, but it degraded for tail-localized signals because the reference sample had too few points where the anomaly lives (Grosso, 2024).

6. Empirical behavior, applications, and limitations

Across comparison studies, NPLM is described as especially effective for small deviations, localized anomalies, and searches that must remain agnostic to the anomaly shape. The key mechanism is the likelihood-ratio weighting: points with large $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$3, corresponding to classifier outputs close to $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$4 or $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$5, contribute more strongly than points near indecision. This suppresses the effect of large background-like regions and enhances localized discrepancies. The reported outcome is a more uniform response across benchmark problems than classifier-based methods or binning-based tests that may be strong for one class of deviations and blind to another (Grosso et al., 2023).

Collider applications remain the primary domain. The literature reports toy problems, a five-dimensional dimuon benchmark, the LHC Olympics dijet anomaly benchmark, a 24-dimensional CMS L1 trigger-like dataset, and resonant anomaly-detection studies. In the resonant setting, NPLM-based methods are reported to outperform BDT-based approaches in detection performance, particularly in low signal injection scenarios, while significantly reducing epistemic variance due to hyperparameter choices. When accurate background modelling is available, an end-to-end NPLM test is preferred; when it is unavailable, NPLM has been used as the anomaly selector in conjunction with a hyper-test over multiple thresholds (Grosso et al., 3 Jan 2025).

Outside discovery searches, NPLM has been repurposed as a validation method for high-dimensional generative models. In that context it is used as a two-sample goodness-of-fit test between generated data and a reference sample, with benchmarks including mixtures of Gaussians of dimensions $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$6 and the 16-observable FlashSim jet generator. The reported findings were that the method tracks model quality, becomes more sensitive as sample size increases, and can diagnose sub-optimally modeled regions through the learned log-density ratio and event-wise anomaly scores (Cappelli et al., 12 Nov 2025, Grossi et al., 4 Aug 2025).

The method also has clear limitations. It requires a sufficiently large and accurate reference sample; if the reference sample is too small, specifically when $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$7, the null distribution of the test statistic can deviate from the expected $$t_{\rm lr}=-2\left[\sum_{x\in R}\left(e^{f(x)}-1\right)-\sum_{x\in D}f(x)\right],$$8 behavior. Hyperparameter tuning remains heuristic, null calibration is computationally expensive because each toy requires retraining, and high-dimensional performance remains constrained by sparsity and the curse of dimensionality. In the batched and streaming work, performance depended on the quality and size of the reference sample, on the choice of input features, and on zero-padding artifacts in variable-size event representations; systematic uncertainties were not handled in that work (Letizia et al., 2022, Grosso, 2024).

A final point of clarification concerns scope. A 2025 satirical paper about searching arXiv abstracts for “New Physics” keywords was explicitly framed as parody and was not a technical NPLM implementation, even though it mimicked the logic of signal-versus-background language and hypothesis testing. That paper is relevant chiefly as a caution against conflating the general rhetoric of machine-learning-based anomaly detection with the specific likelihood-ratio construction that defines NPLM (Gambhir, 28 Mar 2025).