---
title: 'BumpNet: Resonance Searches & PDE Solutions'
url: https://www.emergentmind.com/topics/bumpnet
type: topic
---

# BumpNet: Resonance Searches & PDE Solutions

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) [2509.16282] [2501.05603]. 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 [2512.17198].

## 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 [2509.16282].

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 [2501.05603].

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 \(pp\) collisions at \(\sqrt s=13\) TeV, from which exclusive final states are built by all combinations of at least two reconstructed objects, yielding 39 768 distinct invariant-mass histograms [2509.16282]. Variable binning is chosen to approximate the experimental mass resolution,
\[
\Delta m(m)\;\approx\;\kappa\;\sigma_{\rm res}(m)\quad(\kappa\sim1),
\]
so that a narrow resonance always spans \(\mathcal O(1)\) bin.

The local target significance is computed from known injected signal and background expectations. One reported definition uses the Cowan et al. asymptotic formula
\[
s_i \equiv Z_{\rm local}(i)
= \sqrt{2\Bigl[
(b_i+s_i)\ln\frac{b_i+s_i}{b_i} - s_i
\Bigr]},
\]
with \(b_i\) the known background expectation and \(s_i\) the injected signal in bin \(i\) [2509.16282]. 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 \((n_{\rm bins},1)\) containing the raw Poisson-fluctuated counts \(n_i\), with histogram lengths \(n_{\rm bins}\ge 30\) and a global linear rescaling of \(n_i\) to \([0,1]\) before entering the network [2501.05603]. Another description states that no explicit mass values are passed; only the counts \(\{N_i\}_{i=1}^{N_{\rm bins}}\) are provided [2509.16282].

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 \(K=\{3,5,9,15\}\), \(F\) feature maps with typical \(F\sim32\), stride \(=1\), padding to preserve length, and ReLU activation; the concatenated output has dimension \(N_{\rm bins}\times (4F)\), followed by a bin-wise MLP with two hidden layers of width \(H\) and typical \(H\sim64\), and a final linear output unit \(\hat s_i\in\mathbb R\) [2509.16282]. Another version describes four parallel 1D-convolutional stacks, each with four Conv1D layers of 64 channels and ReLU activations, kernel sizes \(K_k\in\{3,9,15,25\}\), “same” padding, concatenation to \((n_{\rm bins},4\times64)\), a skip-connection with the original input, and an MLP head with layers \(4\times64+1\to128\to64\to32\to1\) [2501.05603]. 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 \(\sim\!3\) million histograms, with \(1/3\) drawn from analytical backgrounds and \(2/3\) from smoothed Monte Carlo backgrounds [2509.16282]. 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 [2509.16282] [2501.05603]. 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 [2509.16282].

Signals are injected as a single-bin-width Gaussian bump at a random position \(m_0\) with random strength \(S\), after which each bin is Poisson-fluctuated according to \(N_i\sim\mathrm{Pois}(b_i+s_i)\) [2509.16282]. Optimization uses mean-squared error between predicted and true significance values, Adam, an initial learning rate of \(\sim10^{-3}\), and early stopping on validation loss or validation MSE [2509.16282] [2501.05603]. One version reports a batch size of 5 000 histograms and a CosineAnnealingLR schedule decaying from \(10^{-3}\) to 0 over 300 epochs [2501.05603]; another reports typical batch size \(\mathcal O(10^2\text{–}10^3)\), learning-rate decay on plateau, and training for tens of epochs [2509.16282].

The network output is linked back to standard hypothesis testing. Under the background-only hypothesis, \(\hat s_i\) is reported to be approximately standard normal for each bin, so that the one-sided local \(p\)-value is
\[
p_{\rm local}(i)=1-\Phi(\hat s_i),
\]
and global significance is obtained by constructing an empirical background-only ensemble of \(\max_i \hat s_i\), scanning both over bins and over histograms, and converting the observed maximum into a global \(p\)-value [2509.16282].

## 4. Collider validation, deployment, and limitations

The collider BumpNet is evaluated against idealized likelihood-ratio-based tests. For injected Gaussians, one reported metric is
\[
\Delta Z_{\max}=Z_{\max}^{\rm BumpNet}-Z_{\max}^{\rm LR}.
\]
The network is described as unbiased, with \(\langle\Delta Z_{\max}\rangle\approx0\) for signals of varying strength, and with spread \(\sigma(\Delta Z_{\max})\lesssim0.3\) over most of the histogram range; excluding the lowest \(10\%\) of bins improves performance to \(\sigma\sim0.2\) [2509.16282]. A separate evaluation reports mean \(\Delta Z_{\max}\simeq -0.09\,\sigma\) and standard deviation \(\simeq0.53\,\sigma\) on held-out analytical-function histograms, and mean \(\Delta Z_{\max}\simeq -0.11\,\sigma\) with standard deviation \(\simeq0.75\,\sigma\) on held-out Dark-Machines histograms, together with positional accuracy at the level of \(\lesssim0.15\) bins and ROC AUC \(\simeq0.88\text{–}0.90\) relative to the ideal likelihood-ratio test [2501.05603]. These are different reported validation settings rather than a single unified benchmark.

Applications to more realistic signals are also reported. For Higgs \(\to\gamma\gamma\) diphoton data, BumpNet gives \(Z_{\max}^{\rm pred}=4.5\sigma\) versus a likelihood-ratio value of \(4.2\sigma\) [2509.16282]. For scalar leptoquark pair production with \(m=600\) GeV injected on Dark-Machines background, a clear BumpNet peak appears at the correct mass [2509.16282]. Another study reports tests on \(O(10)\) simulated BSM signals, including pair-produced scalar leptoquarks, low-mass \(Z'\to\mu\mu\), RPV stops at \(1\) TeV, and \(W'(1.5\) TeV\()\to WZ\), with \(Z_{\rm pred}^{\max}\gg50\,\sigma\) at the correct \(m_{\rm peak}\); wider \(2\)- to \(3\)-bin signals show a modest under-prediction of \(Z\) with \(\Delta Z\sim1\,\sigma\), but remain easily discoverable [2501.05603].

The look-elsewhere effect is treated explicitly. One reported mitigation excludes the first \(10\%\) of bins, where edge effects are strongest, and applies a Global Analysis Algorithm (GAA): seed histograms are identified with \(Z_{\rm pred}^{\max}>Z_{\rm thresh}\) in half of the data, then validated in the other half by requiring a matching excess within \(\pm2\) 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 [2501.05603]. A related description constructs the background-only distribution of the histogram maximum across all scanned bins and histograms [2509.16282].

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 \(\mathcal O(1)\) ms on a GPU [2509.16282]. Training on \(\sim3\times10^6\) histograms is reported to require \(\mathcal O(10^2)\) GPU-hours on multi-GPU clusters, while deployment reaches \(\sim10^3\) histograms/s on a single GPU and \(\sim10^5\) histograms in a few minutes on a small GPU farm, enabling fully unsupervised scanning of \(>39\)k histograms from real data [2509.16282].

The limitations are stated plainly. Known issues include ambiguity in the first \(\sim10\%\) 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 [2509.16282]. 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 [2509.16282]. 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 [2501.05603].

## 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 [2512.17198]. The approximation of a target scalar field \(u(\mathbf{x})\) is written as
\[
\hat u(\mathbf{x};\theta)=\sum_{i=1}^m h^i\,\psi^i(\mathbf{x}),
\]
where \(h^i\in\mathbb R\) is the amplitude of the \(i\)-th bump and \(\psi^i(\mathbf{x})\colon\mathbb R^n\to[0,1]\) is a smooth, compact-support bump function [2512.17198].

Each bump is generated by a two-layer, weight-tied subnetwork. In \(n\) dimensions,
\[
\psi^i(\mathbf{x})
=
\operatorname{squash}\Biggl(
\sum_{j=1}^n \tanh\!\bigl(p^i(\beta_j^{i\,T}\mathbf{x}+s^i_j)\bigr)
+
\sum_{j=1}^n \tanh\!\bigl(p^i(-\beta_j^{i\,T}\mathbf{x}+\bar s^i_j)\bigr)
-(2n-1)
\Biggr),
\]
with
\[
\operatorname{squash}(z)=\tfrac12\bigl(1+\tanh(p^i z)\bigr)\in[0,1].
\]
The inner \(\tanh\) terms define half-spaces whose intersection is an axis-aligned box rotated by the choice of \(\{\beta^i_j\}\), and the outer \(\tanh\) layer smoothly maps the result back into \([0,1]\) [2512.17198].

The parameters admit a geometric interpretation. The center \(\mathbf c^i\), support widths \(l^i_j\), orientation, and sharpness \(p^i\) can be read off from \(\{p^i,\beta^i_j,s^i_j,\bar s^i_j\}\). The paper gives
\[
[\,c^i_1,\dots,c^i_n\,]
=
(B^i)^{-1}
[\,\bar s^i_1-s^i_1,\dots,\bar s^i_n-s^i_n\,],
\qquad
B^i=[\beta^i_1,\ldots,\beta^i_n],
\]
and
\[
l^i_j = \frac{s^i_j+\bar s^i_j}{\|\beta^i_j\|}>0.
\]
To ensure positive widths and keep centers inside a known bounding box \(\prod_{j=1}^n [x_{j\ell},x_{jr}]\), the biases are reparameterized through trainable \(W^i_j\in\mathbb R\) and \(v^i_j\in\mathbb R\), with
\[
l^i_j=e^{W^i_j},
\qquad
c^i_j=x_{j\ell}+\tfrac12\bigl(\tanh(v^i_j)+1\bigr)(x_{jr}-x_{j\ell}) .
\]
This makes the representation directly interpretable in terms of location, size, orientation, and amplitude [2512.17198].

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 [2512.17198]. Dynamic pruning is central to model parsimony and \(h\)-adaptivity: every \(r\) training steps, bumps with amplitude \(|h^i|\) below a threshold \(q\) are permanently removed, concentrating representation in regions of strong gradients and reducing parameter count [2512.17198].

## 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 [2512.17198]. The reported metrics include relative \(L^1\) or \(L^2\) error, trainable parameters, and wall-clock training time.

| Benchmark | BumpNet-family result | Comparator |
|---|---|---|
| Helmholtz, \(70\)K epochs | Bump-PINN: error \(2.66\times10^{-2}\), params \(1{,}575\), time \(51\) s | PINN: error \(1.24\times10^{-1}\), params \(83{,}073\), time \(85\) s |
| 1D Heat, \(20\)K epochs | Bump-PINN: error \(1.83\times10^{-3}\), params \(840\), time \(12\) s | PINN: error \(3.15\times10^{-2}\), params \(3{,}421\), time \(41\) s |
| Advection | Bump-PINN: error \(2.3\times10^{-3}\), params \(154\), time \(28\) s | PINN: error \(7.78\times10^{-2}\), params \(83{,}073\), time \(48\) s |

For time-dependent evolution, Bump-EDNN is reported with \(252\) trainable parameters, \(36\) evolving parameters, final \(L^2\) error \(5\times10^{-4}\), and time-evolution cost \(\approx 6\) s on 2D Heat. The comparison EDNN with an MLP \(4\times20\) uses \(1{,}341\) parameters, evolves all of them, reaches final error \(1.4\times10^{-3}\), and has time-evolution cost \(\approx 23\) min [2512.17198]. For operator learning on the diffusion-reaction problem, the Bump-DeepONet trunk has \(600\) parameters, trains in \(86\) s, and reaches test error \(8.12\times10^{-6}\), whereas the DeepONet trunk has \(25{,}600\) parameters, trains in \(158\) s, and reaches test error \(5.42\times10^{-6}\) [2512.17198].

The stated advantages are parameter efficiency, computational speed, interpretability, and \(h\)-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 [2512.17198]. The limitations are equally specific: the current bump construction assumes a hyperrectangular domain or bounding box; in very high dimension, the \(2n\) half-space neurons per bump and Gram–Schmidt orientation construction may become costly; the pruning hyperparameters \((q,r)\) require tuning; and all experiments are mesh-free collocation, so integration with variational formulations or weak forms remains open [2512.17198].

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.

Source: https://www.emergentmind.com/topics/bumpnet