BumpNet: Resonance Searches & PDE Solutions

• BumpNet is a dual-use neural framework that employs localized “bump” concepts to address both collider resonance searches and PDE operator learning.
• In collider physics, it maps invariant-mass histograms to per-bin local significance values using convolutional layers and MLPs, enabling rapid automated bump hunting at the LHC.
• For PDE solutions, it uses a sparse, interpretable bump function expansion with trainable parameters to efficiently approximate scalar fields and operators.

BumpNet is a name used in the 2025 arXiv literature for distinct neural-network frameworks. In collider physics, BumpNet denotes a fully supervised, end-to-end neural approach to model-agnostic bump-hunting in invariant-mass spectra, trained to map smoothly falling invariant-mass histogram data directly to bin-by-bin local significance values for automated resonance searches at the Large Hadron Collider (LHC) (Arguin et al., 18 Sep 2025, Arguin et al., 9 Jan 2025). In scientific machine learning, BumpNet denotes a sparse neural network framework for PDE numerical solution and operator learning, based on meshless basis function expansion using bump functions constructed from ordinary sigmoid activation functions (Chiu et al., 19 Dec 2025).

1. Terminological scope and disambiguation

The shared name masks two technically unrelated uses.

BumpNet usage	Domain	Core construction
Automated bump hunts	LHC resonance searches	Neural mapping from histogram counts to local significance values
Sparse PDE learning	Scientific ML for PDEs	Weighted sum of localized bump basis functions

In the collider-physics usage, the central object is an invariant-mass histogram built from collider data, and the network output is a predicted local significance for each mass bin. In the PDE usage, the central object is an approximation of a scalar field $u(\mathbf{x})$ by a sparse sum of localized basis functions whose shape, location, and amplitude are trainable.

A common source of confusion is therefore nominal rather than methodological: the two BumpNet lines share neither application domain nor architectural objective. This suggests that references to “BumpNet” require field-specific qualification, especially in bibliographic or survey contexts.

2. BumpNet for automated resonance searches

The collider-physics BumpNet was introduced in the context of the absence of an unambiguous Beyond-Standard-Model (BSM) resonance at the LHC, despite increasing energy and luminosity. The stated motivation is that traditional bump-hunting techniques typically rely on choosing a parametric form for the smoothly falling background in each final state, fitting signal plus background or scanning sliding windows with ad-hoc sideband definitions, and manually inspecting $O(10)$ channels, whereas LHC data encompass thousands of exclusive final states such as every possible pair or triplet of electrons, muons, jets, photons, $E_T^{\rm miss}$, and boosted $W/Z$ candidates (Arguin et al., 18 Sep 2025).

The method is framed as a direct replacement for repeated per-channel likelihood scans. Under Poisson statistics, one writes $\mu_i^{(B)} = B_i$ for the background-only hypothesis and $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$ for the signal-plus-background hypothesis. Conventional analyses scan a signal location $m_0$, compute a profile-likelihood ratio, and convert it into a local significance $Z(m_0)$. BumpNet instead learns the direct mapping

$\{n_i\} \to \{Z(m_i)\},$

so that the full histogram is processed in one pass rather than through repeated optimizer calls at each candidate mass (Arguin et al., 9 Jan 2025).

The training and evaluation pipeline is built on LHC-like histograms. One starting point is the Dark-Machines Monte Carlo dataset, described as $\sim\!10\;\mathrm{fb}^{-1}$ of Standard Model $O(10)$0 collisions at $O(10)$1 TeV, from which exclusive final states are built by all combinations of at least two reconstructed objects, yielding 39 768 distinct invariant-mass histograms (Arguin et al., 18 Sep 2025). Variable binning is chosen to approximate the experimental mass resolution,

$O(10)$2

so that a narrow resonance always spans $O(10)$3 bin.

The local target significance is computed from known injected signal and background expectations. One reported definition uses the Cowan et al. asymptotic formula

$O(10)$4

with $O(10)$5 the known background expectation and $O(10)$6 the injected signal in bin $O(10)$7 (Arguin et al., 18 Sep 2025). This establishes BumpNet as a supervised significance-regression model rather than an unsupervised anomaly detector.

3. Collider architecture, training data, and statistical mapping

The collider BumpNet takes as input a one-dimensional vector of bin counts. One description specifies a tensor of shape $O(10)$8 containing the raw Poisson-fluctuated counts $O(10)$9, with histogram lengths $E_T^{\rm miss}$0 and a global linear rescaling of $E_T^{\rm miss}$1 to $E_T^{\rm miss}$2 before entering the network (Arguin et al., 9 Jan 2025). Another description states that no explicit mass values are passed; only the counts $E_T^{\rm miss}$3 are provided (Arguin et al., 18 Sep 2025).

The arXiv record contains closely related but not identical architectural descriptions. One version uses four parallel stacks, each consisting of a 1D convolutional layer with kernel sizes such as $E_T^{\rm miss}$4, $E_T^{\rm miss}$5 feature maps with typical $E_T^{\rm miss}$6, stride $E_T^{\rm miss}$7, padding to preserve length, and ReLU activation; the concatenated output has dimension $E_T^{\rm miss}$8, followed by a bin-wise MLP with two hidden layers of width $E_T^{\rm miss}$9 and typical $W/Z$0, and a final linear output unit $W/Z$1 (Arguin et al., 18 Sep 2025). Another version describes four parallel 1D-convolutional stacks, each with four Conv1D layers of 64 channels and ReLU activations, kernel sizes $W/Z$2, “same” padding, concatenation to $W/Z$3, a skip-connection with the original input, and an MLP head with layers $W/Z$4 (Arguin et al., 9 Jan 2025). This suggests architectural refinement between arXiv records while preserving the same operational objective: per-bin regression of local significance.

The training dataset is reported as $W/Z$5 million histograms, with $W/Z$6 drawn from analytical backgrounds and $W/Z$7 from smoothed Monte Carlo backgrounds (Arguin et al., 18 Sep 2025). The analytical component comprises eleven families of monotonic, concave or smoothly falling functions, including power laws, exponentials, and log-polynomial forms, with parameter sampling chosen to guarantee a monotonically falling distribution over the histogram range or to enlarge shape diversity (Arguin et al., 18 Sep 2025, Arguin et al., 9 Jan 2025). The Monte Carlo-derived component uses Dark-Machines histograms fit with parametric or non-parametric smoothing techniques, ensuring realistic shapes that include detector acceptance and kinematic thresholds (Arguin et al., 18 Sep 2025).

Signals are injected as a single-bin-width Gaussian bump at a random position $W/Z$8 with random strength $W/Z$9, after which each bin is Poisson-fluctuated according to $\mu_i^{(B)} = B_i$0 (Arguin et al., 18 Sep 2025). Optimization uses mean-squared error between predicted and true significance values, Adam, an initial learning rate of $\mu_i^{(B)} = B_i$1, and early stopping on validation loss or validation MSE (Arguin et al., 18 Sep 2025, Arguin et al., 9 Jan 2025). One version reports a batch size of 5 000 histograms and a CosineAnnealingLR schedule decaying from $\mu_i^{(B)} = B_i$2 to 0 over 300 epochs (Arguin et al., 9 Jan 2025); another reports typical batch size $\mu_i^{(B)} = B_i$3, learning-rate decay on plateau, and training for tens of epochs (Arguin et al., 18 Sep 2025).

The network output is linked back to standard hypothesis testing. Under the background-only hypothesis, $\mu_i^{(B)} = B_i$4 is reported to be approximately standard normal for each bin, so that the one-sided local $\mu_i^{(B)} = B_i$5-value is

$\mu_i^{(B)} = B_i$6

and global significance is obtained by constructing an empirical background-only ensemble of $\mu_i^{(B)} = B_i$7, scanning both over bins and over histograms, and converting the observed maximum into a global $\mu_i^{(B)} = B_i$8-value (Arguin et al., 18 Sep 2025).

4. Collider validation, deployment, and limitations

The collider BumpNet is evaluated against idealized likelihood-ratio-based tests. For injected Gaussians, one reported metric is

$\mu_i^{(B)} = B_i$9

The network is described as unbiased, with $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$0 for signals of varying strength, and with spread $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$1 over most of the histogram range; excluding the lowest $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$2 of bins improves performance to $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$3 (Arguin et al., 18 Sep 2025). A separate evaluation reports mean $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$4 and standard deviation $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$5 on held-out analytical-function histograms, and mean $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$6 with standard deviation $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$7 on held-out Dark-Machines histograms, together with positional accuracy at the level of $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$8 bins and ROC AUC $\mu_i^{(S+B)}(\theta)=B_i+\theta S_i$9 relative to the ideal likelihood-ratio test (Arguin et al., 9 Jan 2025). These are different reported validation settings rather than a single unified benchmark.

Applications to more realistic signals are also reported. For Higgs $m_0$0 diphoton data, BumpNet gives $m_0$1 versus a likelihood-ratio value of $m_0$2 (Arguin et al., 18 Sep 2025). For scalar leptoquark pair production with $m_0$3 GeV injected on Dark-Machines background, a clear BumpNet peak appears at the correct mass (Arguin et al., 18 Sep 2025). Another study reports tests on $m_0$4 simulated BSM signals, including pair-produced scalar leptoquarks, low-mass $m_0$5, RPV stops at $m_0$6 TeV, and $m_0$7 TeV$m_0$8, with $m_0$9 at the correct $Z(m_0)$0; wider $Z(m_0)$1- to $Z(m_0)$2-bin signals show a modest under-prediction of $Z(m_0)$3 with $Z(m_0)$4, but remain easily discoverable (Arguin et al., 9 Jan 2025).

The look-elsewhere effect is treated explicitly. One reported mitigation excludes the first $Z(m_0)$5 of bins, where edge effects are strongest, and applies a Global Analysis Algorithm (GAA): seed histograms are identified with $Z(m_0)$6 in half of the data, then validated in the other half by requiring a matching excess within $Z(m_0)$7 bins at the same mass and the same object combination, with histograms dropped if they fail to form a family of at least two uncorrelated excesses (Arguin et al., 9 Jan 2025). A related description constructs the background-only distribution of the histogram maximum across all scanned bins and histograms (Arguin et al., 18 Sep 2025).

The practical motivation is scalability. Sideband or sliding-window tests require a separate fit per window and manual configuration, whereas BumpNet processes an entire histogram in one forward pass, reported as $Z(m_0)$8 ms on a GPU (Arguin et al., 18 Sep 2025). Training on $Z(m_0)$9 histograms is reported to require $\{n_i\} \to \{Z(m_i)\},$0 GPU-hours on multi-GPU clusters, while deployment reaches $\{n_i\} \to \{Z(m_i)\},$1 histograms/s on a single GPU and $\{n_i\} \to \{Z(m_i)\},$2 histograms in a few minutes on a small GPU farm, enabling fully unsupervised scanning of $\{n_i\} \to \{Z(m_i)\},$3k histograms from real data (Arguin et al., 18 Sep 2025).

The limitations are stated plainly. Known issues include ambiguity in the first $\{n_i\} \to \{Z(m_i)\},$4 of bins, dependence on binning choices when bin widths vary wildly, and the absence of systematic uncertainties such as detector effects and background mismodeling from the training set (Arguin et al., 18 Sep 2025). Related future directions include variable-width convolutions or learnable kernel sizes, systematic-variation histograms in training, extension to two-dimensional mass distributions or angular-mass joint searches, and a fully automated global-significance pipeline across all channels (Arguin et al., 18 Sep 2025). A separate outlook emphasizes better modeling of broad signals, improved treatment of sparse-statistics high-mass bins, integration of the smoothing or fitting step into the network, and extension to two-dimensional or event-based anomaly detection via encoders plus BumpNet (Arguin et al., 9 Jan 2025).

5. BumpNet as a sparse framework for PDE solution and operator learning

A separate BumpNet introduces a sparse neural network framework for PDE numerical solution and operator learning, based on meshless basis function expansion in a fashion similar to radial-basis function networks, but using basis functions constructed from ordinary sigmoid activation functions (Chiu et al., 19 Dec 2025). The approximation of a target scalar field $\{n_i\} \to \{Z(m_i)\},$5 is written as

$\{n_i\} \to \{Z(m_i)\},$6

where $\{n_i\} \to \{Z(m_i)\},$7 is the amplitude of the $\{n_i\} \to \{Z(m_i)\},$8-th bump and $\{n_i\} \to \{Z(m_i)\},$9 is a smooth, compact-support bump function (Chiu et al., 19 Dec 2025).

Each bump is generated by a two-layer, weight-tied subnetwork. In $\sim\!10\;\mathrm{fb}^{-1}$0 dimensions,

$\sim\!10\;\mathrm{fb}^{-1}$1

with

$\sim\!10\;\mathrm{fb}^{-1}$2

The inner $\sim\!10\;\mathrm{fb}^{-1}$3 terms define half-spaces whose intersection is an axis-aligned box rotated by the choice of $\sim\!10\;\mathrm{fb}^{-1}$4, and the outer $\sim\!10\;\mathrm{fb}^{-1}$5 layer smoothly maps the result back into $\sim\!10\;\mathrm{fb}^{-1}$6 (Chiu et al., 19 Dec 2025).

The parameters admit a geometric interpretation. The center $\sim\!10\;\mathrm{fb}^{-1}$7, support widths $\sim\!10\;\mathrm{fb}^{-1}$8, orientation, and sharpness $\sim\!10\;\mathrm{fb}^{-1}$9 can be read off from $O(10)$00. The paper gives

$O(10)$01

and

$O(10)$02

To ensure positive widths and keep centers inside a known bounding box $O(10)$03, the biases are reparameterized through trainable $O(10)$04 and $O(10)$05, with

$O(10)$06

This makes the representation directly interpretable in terms of location, size, orientation, and amplitude (Chiu et al., 19 Dec 2025).

The framework is instantiated in several variants. Bump-PINNs combine BumpNet with physics-informed neural networks and use the same collocation-based approach used by PINNs; Bump-SAPINN introduces self-adaptive weights following McClenny and Braga-Neto (2023); Bump-EDNN uses a BumpNet only in the spatial domain and EDNNs to advance the solution in time; Bump-DeepONets employ a BumpNet regression network as the trunk network of a DeepONet (Chiu et al., 19 Dec 2025). Dynamic pruning is central to model parsimony and $O(10)$07-adaptivity: every $O(10)$08 training steps, bumps with amplitude $O(10)$09 below a threshold $O(10)$10 are permanently removed, concentrating representation in regions of strong gradients and reducing parameter count (Chiu et al., 19 Dec 2025).

6. PDE benchmarks, comparative results, and open questions

The PDE-oriented BumpNet is evaluated on inhomogeneous Helmholtz, 2D Poisson, 1D Heat, 1D Advection, 2D Heat, and nonlinear diffusion-reaction operator learning, with comparisons to PINN, SAPINN, SPINN of Ramabathiran and Ramachandran, EDNN, and DeepONet (Chiu et al., 19 Dec 2025). The reported metrics include relative $O(10)$11 or $O(10)$12 error, trainable parameters, and wall-clock training time.

Benchmark	BumpNet-family result	Comparator
Helmholtz, $O(10)$13K epochs	Bump-PINN: error $O(10)$14, params $O(10)$15, time $O(10)$16 s	PINN: error $O(10)$17, params $O(10)$18, time $O(10)$19 s
1D Heat, $O(10)$20K epochs	Bump-PINN: error $O(10)$21, params $O(10)$22, time $O(10)$23 s	PINN: error $O(10)$24, params $O(10)$25, time $O(10)$26 s
Advection	Bump-PINN: error $O(10)$27, params $O(10)$28, time $O(10)$29 s	PINN: error $O(10)$30, params $O(10)$31, time $O(10)$32 s

For time-dependent evolution, Bump-EDNN is reported with $O(10)$33 trainable parameters, $O(10)$34 evolving parameters, final $O(10)$35 error $O(10)$36, and time-evolution cost $O(10)$37 s on 2D Heat. The comparison EDNN with an MLP $O(10)$38 uses $O(10)$39 parameters, evolves all of them, reaches final error $O(10)$40, and has time-evolution cost $O(10)$41 min (Chiu et al., 19 Dec 2025). For operator learning on the diffusion-reaction problem, the Bump-DeepONet trunk has $O(10)$42 parameters, trains in $O(10)$43 s, and reaches test error $O(10)$44, whereas the DeepONet trunk has $O(10)$45 parameters, trains in $O(10)$46 s, and reaches test error $O(10)$47 (Chiu et al., 19 Dec 2025).

The stated advantages are parameter efficiency, computational speed, interpretability, and $O(10)$48-adaptivity. The framework often uses an order of magnitude or more fewer parameters than PINN or DeepONet baselines, while preserving a direct interpretation of each bump’s center, support, orientation, and height (Chiu et al., 19 Dec 2025). The limitations are equally specific: the current bump construction assumes a hyperrectangular domain or bounding box; in very high dimension, the $O(10)$49 half-space neurons per bump and Gram–Schmidt orientation construction may become costly; the pruning hyperparameters $O(10)$50 require tuning; and all experiments are mesh-free collocation, so integration with variational formulations or weak forms remains open (Chiu et al., 19 Dec 2025).

Taken together, the two BumpNet lineages occupy different parts of the machine-learning landscape. One learns a likelihood-ratio-like significance map for large-scale resonance searches over invariant-mass histograms; the other provides a sparse, interpretable basis-function architecture for PDE solution and operator learning. The shared name reflects a common intuition of localized “bumps,” but the underlying statistical objects, inductive biases, and deployment regimes are distinct.