Modular Jets Across Disciplines

Updated 12 December 2025

Modular jets are systems composed of discrete, interacting modules, applied in fields from astrophysics to machine learning to disentangle complex behaviors.
In astrophysics, modular jets reveal multiple kinematic streams in YSO outflows, challenging traditional rotation models through discrete velocity gradients.
Modular frameworks in particle physics and engineering, as well as diagnostics in learning pipelines, enable systematic uncertainty quantification and optimized component integration.

A modular jet is defined as a physical or mathematical jet-like outflow, structure, or computational system exhibiting composition from discrete, interacting modules or sub-units. Modularity in jets arises across a range of disciplines: in the astrophysical context, it refers to multi-component outflows distinguished by geometric or kinematic segmentation; in particle and nuclear physics, it denotes plug-and-play theoretical frameworks where distinct modules capture different physical regimes; in machine learning, the term formalizes locality and response properties of modular neural or regression pipelines. Modular jets enable refined modeling of complex behaviors, analysis of internal structure, and systematic studies of system identifiability or ambiguity.

1. Modular Jets in Astrophysics

Observations of young stellar object (YSO) outflows have revealed velocity gradients and multi-peaked features that challenge simple rotation-based explanations. In NGC 1333 IRAS 4A2, high–angular–resolution SiO observations show systematic velocity gradients across jet cuts: the sense of gradient remains fixed across both lobes, and the position–velocity diagrams display up to four discrete intensity peaks per cut, inconsistent with solid-body rotation. Solid-body rotation would require opposite Doppler signatures in the two lobes and yield a single, smoothly varying velocity across the jet. Derived jet-specific angular momentum and mass-loss rates in the disk–wind framework are incompatible with expected lever-arm values and observed jet outflows, rendering the rotation scenario physically untenable (Soker et al., 2012).

Instead, a modular decomposition is required: the outflow is modeled as two oppositely directed sub-jets, each composed of two collimated components, for four discrete streams. Each component exhibits a distinct inclination ( $\theta$ ) and azimuthal ( $\phi$ ) angle, but all components flow at the same bulk velocity. The geometric axis of the modular jet structure remains invariant over $\sim$ 500 yr, matching a fixed plane in the launching accretion disk, and the preferential directions are attributed to disk perturbations induced by a companion in a highly eccentric orbit. At each periastron, tidal excitation of an $m=2$ perturbation produces spiral arms that seed episodic and directionally fixed modular jets. This paradigm robustly predicts periodic jet activity and synchronous sub-jet brightening with invariant position angles, distinguishing modular jets from monolithic precessing or rotating interpretations.

2. Modular Jets in Theoretical and Computational Particle Physics

In jet quenching and jet-medium interaction studies, the modular approach structures simulation frameworks into sequential composable modules, each addressing a distinct physical regime. The JETSCAPE framework exemplifies this: high virtuality ( $Q^2 \gg Q^2_{sw}$ ) parton evolution is handled by the MATTER module, incorporating DGLAP-like showering and medium-induced effects, while a switch occurs at $Q^2_{sw}$ to modules suitable for low virtuality, such as LBT (Linear Boltzmann Transport), MARTINI (AMY-based), or AdS/CFT drag modules (Priyadarshini et al., 2023).

Each module is calibrated to a specific regime: MATTER applies perturbative QCD with virtuality-ordered emissions and medium modifications; LBT and MARTINI model elastic and inelastic interactions at lower virtuality, employing different recoil and energy-absorption schemes; AdS/CFT provides a holographic strong-coupling drag description absent explicit splitting. Jet observables—such as the nuclear modification factor $R_{AA}(R, p_T)$ , its double ratio across jet cone radii, and jet substructure measures—are computed by plugging these modules together, enabling systematic comparison to LHC data across an extended $p_T$ and angular range, and facilitating isolatable studies of module-specific uncertainties and physical assumptions. This modular strategy supports substitutability, scalability, and transparent physical interpretation.

3. Modular Jets in Machine Learning Pipelines

The modular jet formalism has been advanced for diagnosing the internal structure, identifiability, and mirage regimes in deep learning pipelines (Sanyal, 5 Dec 2025). Here, a modular pipeline is viewed as a composition of $K$ modules,

$x \xrightarrow{M_1} z_1 \xrightarrow{M_2} z_2 \rightarrow \cdots \rightarrow z_K = \hat y,$

with each module $M_m$ exposing an intermediate representation. The empirical modular jet at base input $x_0$ for module $m$ is defined as the pair $(z_m(x_0), A_m(x_0))$ , where $A_m$ is the local Jacobian (linear response) of $z_m$ with respect to $x$ .

Empirical jets are estimated by perturbing $x_0$ and regressing finite-difference responses, thereby quantifying each module's sensitivity structure. Modular jets serve two critical roles: they enable diagnosis of “mirage” regimes (where multiple distinct pipeline decompositions yield identical output and indistinguishable module-level jets) and define criteria for identifiability (where observed jets uniquely determine internal factorization up to natural symmetries). A two-module linear regression jet-identifiability theorem rigorously formalizes this distinction: risk-only evaluation admits an extensive family of observationally equivalent decompositions; inclusion of module-level jets collapses this ambiguity.

The MoJet algorithm operationalizes empirical jet computation and provides diagnostics (e.g., numerical rank, jet similarity), augmenting risk-based evaluation with fine-grained geometric characterization of learned representations and their local perturbative behavior.

4. Methodologies and Diagnostic Algorithms

Empirical investigation of modular jets proceeds via systematic perturbation, module-level tapping, and regularized local regression:

For each base input, construct a family of small input perturbations;
Tap module outputs for each perturbed input;
Estimate the local Jacobian via ridge regression,

$A_m(x_0)^\top = (\Delta^\top\Delta+\lambda I)^{-1}\Delta^\top Y_m,$

yielding the empirical jet $J_m(x_0) = (z_m(x_0), A_m(x_0))$ ;

Compute diagnostics: numerical rank of $A_m$ , principal-angle subspace affinity (JetSim $[A, B]$ ).

Identifiability is probed by comparing jets across models or decompositions: high similarity and rank congruence indicate mirage regimes, while disparities identify non-trivial internal differences invisible to risk-based metrics. In practice, pipeline architectures (linear, deep, or hybrid) and classification pipelines (e.g., PCA+logistic vs. MLP) can thus be quantitatively dissected at module level.

5. Modular Integration in Engineering Systems

In propulsion engineering, modular jet concepts arise in the design and integration of turbojet engines within turbine-based combined cycle vehicles. The intake, core, and nozzle are each treated as modules subject to iterative scaling to achieve specific performance targets under geometric and thermodynamic constraints (Rajashankar et al., 27 Jun 2024). The process involves:

Intake module: shock-on-cowl-lip conditions fix geometry; mass flow is set by cowl height and shock positioning;
Core engine module: map scaling adjusts baseline compressor/turbine to target mass flow and shaft work, constrained by temperature limits;
Nozzle module: SERN geometry adapted for both scramjet and turbojet modes; efficiency quantified by actual-to-ideal thrust ratio;
Iterative core mass flow update ensures target thrust is met within system margin constraints.

Such modular engineering approaches facilitate integration in multi-regime vehicles (e.g., TBCCs), enabling coordinated optimization over typically incompatible operating environments.

6. Comparative Context and Broader Implications

Across astrophysics, high-energy physics, machine learning, and engineering, the modular jet approach formalizes decomposition—whether geometric, physical, or algorithmic—to enable diagnosis, modeling fidelity, and identifiability. In astronomy, modular jets support new interpretations of outflow kinematics and origin, contrasting with classical monolithic models. In jet quenching, modular computation enables systematic uncertainty quantification and subsystem intercomparison. In deep learning, modular jets structure the analysis of internal representations and open new regimes of model auditability.

A plausible implication is that modularity serves as a powerful lens for disentangling internal complexity, diagnosing model ambiguity, and guiding the design of both physical and algorithmic systems. In each context, empirical diagnostics—whether velocity gradients, $R_{AA}$ as a function of jet radius, or module-level Jacobians—play a crucial role in discriminating between modular scenarios and mirage indistinguishability.