Catapult Phase across Domains

• Catapult phase is a dynamic interval during which stored mechanical, elastic, or structural energy is rapidly released to drive high-energy emissions or transitions.
• In nuclear fission and atomization, it explains how transient deformations—such as healing neck remnants or vortex-shed droplets—convert energy into directed motion.
• In deep learning, the catapult phase denotes a transient large-learning-rate regime where loss spikes and curvature drops facilitate convergence to flatter minima.

Searching arXiv for recent and foundational papers on “catapult phase” across the domains represented in the provided data, with emphasis on the specified 2026 fission paper and the machine-learning literature where the term is formalized as a training regime. In contemporary scientific literature, “catapult phase” is a domain-specific term rather than a single standardized concept. In nuclear fission it denotes a short post-scission interval in which an inward-healing neck remnant can kinematically boost nucleons into high-energy neutron emission; in atomization it denotes a vortex-mediated droplet-ejection sequence tied to Kelvin–Helmholtz wave evolution; and in optimization it denotes a transient large-learning-rate regime in which loss rises before sharpness decreases and training re-enters a stable region (Randrup et al., 12 Apr 2026, Jerome et al., 2016, Lewkowycz et al., 2020). Taken together, these usages suggest a recurring motif: a brief transition in which stored geometric, elastic, surface, or dynamical structure is converted into a rapid release event.

1. Scope of the term

The term is used in several technically distinct ways.

Domain	Meaning of “catapult phase”	Representative paper
Nuclear fission	short post-scission interval with an inward-moving bulge that can reflect nucleons	(Randrup et al., 12 Apr 2026)
Atomization	coupled Kelvin–Helmholtz wave / recirculation-vortex / liquid-film interaction	(Jerome et al., 2016)
Deep learning	transient large-learning-rate regime with loss spikes and sharpness reduction	(Lewkowycz et al., 2020)
Biomechanics and biology	rapid release stage of elastic or capillary energy into droplet or body motion	(Ha et al., 14 May 2026, Kiss et al., 2024, Iapichino et al., 2019)

In the cited literature, the expression does not denote a universal mathematical object. In some papers it is a sharply defined transient interval with a specific mechanism and threshold structure, while in others it is an analogy for an active release stage. This is especially clear from the contrast between the fission paper, which defines a precise post-scission interval, and the machine-learning literature, where the phrase names a training regime between monotone stability and divergence or other non-monotone phases (Randrup et al., 12 Apr 2026, Chen et al., 2023).

A useful cross-domain reading is therefore inferential rather than terminological. The shared structure is a metastable configuration followed by a release episode: healing neck remnants in fission, shed vortices in wakes, super-critical updates in optimization, elastic loading in locomotion, or capillary coalescence in fungal spore launch. This suggests that “catapult phase” is best understood as a family resemblance term rather than a single theory.

2. Post-scission nuclear fission

In “Catapult neutrons from neck snapping in fission,” the catapult phase is the short post-scission interval in which each freshly separated fission fragment still carries a local bulge at the tip facing the other fragment. As the fragment relaxes toward a smoother equilibrium shape, that bulge moves inward. A nucleon that strikes this moving surface can be reflected like a ball from a moving wall, gain energy, become unbound, and later escape as a high-energy neutron. The paper stresses that this is not ordinary evaporation from an equilibrated fragment, but a direct kinematic boost from the collective motion of the bulge surface itself (Randrup et al., 12 Apr 2026).

The post-scission fragment is modeled as a spheroidal reference surface with a Gaussian bulge,

$$h(s;t)=h_0(t)\,g(s), \qquad g(s)=e^{-s^2/2\sigma_0^2},$$

and

$\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$

with a typical axis ratio $c/a=1.8$. The bulge stores excess surface energy

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$

with nuclear surface tension $\gamma \approx 0.9\ \text{MeV/fm}^2$ for a typical fragment. Its collapse is driven by

$F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$

and damped by one-body wall dissipation,

$$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$

For the standard parameters, the initial central surface speed is about

$\dot h_0(t=0)\approx -0.2\,v_F,$

so the bulge motion is rapid but still slower than the Fermi velocity.

The kinematic boost comes from reflection off an inward-moving surface. If a nucleon arrives with normal velocity $v_\perp$, the paper gives the energy change

$\Delta E=-2mUv_\perp+2mU^2,$

where $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$0 for inward motion. The first term is then positive and usually dominates. A qualitative estimate in the paper is that if the inward surface speed is about $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$1 of the Fermi speed, a nucleon near the Fermi surface can gain on the order of 10 MeV. Escape is a two-step process: first the nucleon must be boosted above

$\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$2

and then it must later reach the fragment surface with sufficient normal energy $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$3. If it escapes, the emitted neutron has reduced normal energy

$\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$4

while the tangential component is unchanged.

The simulations sampled millions of test nucleons from a Fermi-Dirac distribution at $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$5 MeV. A few percent of reflected nucleons become unbound after the first surface reflection, and only about one third of those ultimately escape as catapult neutrons. For standard bulge parameters $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$6 fm and $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$7 fm, the paper gives $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$8 per fragment and mean catapult-neutron energy $\frac{\rho_s^2}{a^2}+\frac{z_s^2}{c^2}=1,$9. For both fragments together this corresponds to roughly $c/a=1.8$0 catapult neutrons per fission event, and for thermal-neutron-induced fission of $c/a=1.8$1U to about $c/a=1.8$2 of the total prompt neutron multiplicity. The spectrum is much harder than evaporation, dominates above about $c/a=1.8$3, and has a hard tail around 10 MeV with an effective temperature of roughly $c/a=1.8$4. A common misconception rejected by the paper is that these neutrons are simply prompt evaporation neutrons; the mechanism is instead a moving nuclear wall converting surface energy into single-particle kinetic energy.

3. Wake-driven atomization

In “Vortices catapult droplets in atomization,” the catapult phase is a coupled Kelvin–Helmholtz wave / recirculation-vortex / liquid-film interaction that ejects droplets at unusually large angles. The mechanism is explicit and sequential: a recirculation appears in the wake of the wave while a liquid film emerges from the wave crest; the recirculation detaches into a vortex and the gas flow over the wave momentarily reattaches, pushing the thin liquid film downward; a new recirculation region then forms exactly where the film is located, blows the film upward from below like a bag breakup event, and the ruptured droplets are catapulted by the shed vortex (Jerome et al., 2016).

The precursor structure is a nonlinear Kelvin–Helmholtz wave that propagates downstream in a self-similar manner. Its dynamics depend mainly on the density ratio

$c/a=1.8$5

with self-similar coordinates

$c/a=1.8$6

and asymptotic vorticity

$c/a=1.8$7

The wave crest stretches into a thin liquid film that flaps: during one flapping cycle it first moves downward and then upward. In the catapult regime, droplet ejection is synchronized with both the film flapping period and the vortex shedding period.

The gas flow over the wave is compared to the wake of a backward-facing step. The wave behaves as an obstacle, the gas separates behind it, a recirculation bubble forms, the bubble sheds, and the flow reattaches. This topological interpretation is important because the paper argues that the ejection is not simply shear stripping. The decisive structure is the timed sequence of vortex shedding, temporary reattachment, and new-vortex formation beneath the film.

The nondimensional control parameters are

$c/a=1.8$8

and the DNS uses $c/a=1.8$9 and $V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$0 so that viscosity and capillarity do not dominate the phenomenon. For the droplet-formation period the paper derives

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$1

after estimating the film thickness as

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$2

This scaling captures the DNS trend that droplet formation becomes slower as the density ratio decreases.

The gas-speed dependence is non-monotonic. At low gas speed the recirculation zone stays attached, droplets tend to fall back or eject at negative angles, and there is no catapult. At intermediate gas speed vortex shedding begins, the full catapult sequence appears, and droplets can be thrown at large positive angles, up to about $V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$3 in the experiments. At very high gas speed the wave becomes smaller and thinner, the film bursts more abruptly, there is not enough time for the synchronized cycle, and the ejection angle decreases again. The paper’s “bag-breakup from below” analogy is therefore precise: the newly formed recirculation vortex acts as the inflating agent beneath the wave crest.

4. Large-learning-rate catapult in deterministic optimization

In deep-learning theory, the catapult phase is a large-learning-rate training regime positioned between the small-learning-rate “lazy” phase and outright divergence. The foundational formulation in “The large learning rate phase of deep learning: the catapult mechanism” states that when the initialization curvature is too high for convergence to a nearby point, optimization begins with a period of exponential growth in the loss, coupled with a rapid decrease in curvature, until curvature stabilizes at a value $V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$4. Training then converges to a flatter minimum than those found in the lazy phase (Lewkowycz et al., 2020).

For the solvable one-hidden-layer linear network with MSE loss,

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$5

the relevant curvature variable is

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$6

The paper identifies three regimes: $V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$7 for the lazy phase,

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$8

for the catapult phase in the solvable model, and

$V_{\rm bulge}(h_0)=\gamma\,\Delta S(h_0),$9

for divergence. A key point is that this phenomenon is non-perturbative in width: the transient lasts for a number of training steps of order $\gamma \approx 0.9\ \text{MeV/fm}^2$0, so the relevant asymptotic regime differs from standard fixed-time NTK limits.

Quadratic approximations sharpen this picture. “Quadratic models for understanding catapult dynamics of neural networks” shows that linearized models cannot exhibit catapult behavior, whereas Neural Quadratic Models can. For a single training example the same threshold structure reappears, with sub-critical, catapult, and divergence windows governed by $\gamma \approx 0.9\ \text{MeV/fm}^2$1 and $\gamma \approx 0.9\ \text{MeV/fm}^2$2, up to finite-width corrections. In this regime the loss first grows, the tangent kernel scalar decreases, and the dynamics later stabilize. The paper argues that quadratic models track both optimization and generalization behavior of finite-width neural networks more faithfully than purely linearized models (Zhu et al., 2022).

“Catapult Dynamics and Phase Transitions in Quadratic Nets” makes the self-stabilizing mechanism explicit in a quadratic-network setting. Its central claim is that for a certain range of super-critical learning rates the weight norm decreases whenever the loss becomes large; this reduces the effective curvature or NTK top eigenvalue and restores stability. The paper characterizes the catapult phase by the sequence of initial rapid loss growth, drop in weight norm, drop in effective NTK top eigenvalue, stabilization, and eventual descent to a low-loss solution. It also reports that ReLU networks trained with larger learning rates exhibit activation sparsification, which further alters the effective geometry (Meltzer et al., 2023).

A recurring misconception in this literature is that catapult dynamics are merely a form of failed optimization. These papers define the regime more narrowly: the system is initially unstable relative to the linearized or local stability threshold, but the instability is self-limiting because the trajectory reshapes its own geometry. Another important point is that the catapult phase is not synonymous with the entire edge-of-stability literature. Some works treat catapult as the early transient that moves the system into a later, more persistent near-threshold regime.

5. Stochastic, momentum, and beyond-EOS refinements

Later work refines the deterministic picture along several axes: discrete-time bifurcation theory, minibatch stochasticity, momentum, warmup, feature learning, and objective-function regularity. In “From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression,” constant-step-size gradient descent reduces to the cubic map

$\gamma \approx 0.9\ \text{MeV/fm}^2$3

with five phases as the step size varies: monotonic, catapult, periodic, chaotic, and divergent. The catapult phase occurs precisely for

$\gamma \approx 0.9\ \text{MeV/fm}^2$4

where $\gamma \approx 0.9\ \text{MeV/fm}^2$5 but is not eventually monotone: the loss converges to zero through repeated spikes rather than smooth contraction (Chen et al., 2023).

Stochastic minibatch training changes the phase diagram. “Large Spikes in Stochastic Gradient Descent: A Large-Deviations View” studies NTK-scaled minibatch SGD and defines the catapult phase as the regime in which SGD can create a large transient growth in $\gamma \approx 0.9\ \text{MeV/fm}^2$6 and, as a consequence, flatten the NTK. The critical minibatch interval is

$\gamma \approx 0.9\ \text{MeV/fm}^2$7

but the decisive bifurcation is the sign of

$\gamma \approx 0.9\ \text{MeV/fm}^2$8

If $\gamma \approx 0.9\ \text{MeV/fm}^2$9, large spikes occur with high probability and flatten the kernel; if $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$0, their probability decays polynomially like $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$1, where $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$2 is characterized by

$F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$3

This result separates an inflationary from a deflationary subregime inside the broader catapult window (Gess et al., 10 Mar 2026).

Momentum modifies both thresholds and phenomenology. “Gradient Descent with Polyak’s Momentum Finds Flatter Minima via Large Catapults” studies

$F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$4

with maximum stable sharpness

$F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$5

The paper defines a large catapult as a sharp increase in loss followed by a rapid reduction in sharpness, and argues that momentum prolongs the self-stabilization effect after the trajectory has crossed the stability boundary. Empirically, PHB exhibits much steeper sharpness drops than ordinary GD and settles at sharpness levels well below MSS (Phunyaphibarn et al., 2023).

Warmup and beyond-EOS training reinterpret catapult as a controllable mechanism. “Why Warmup the Learning Rate? Underlying Mechanisms and Improvements” argues that warmup works mainly because it lets the network tolerate larger target learning rates by forcing it into more well-conditioned areas of the loss landscape. For SGD the instability threshold is approximately $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$6, with late-time momentum threshold behaving like $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$7. The paper distinguishes progressive sharpening, sharpness reduction, and constant sharpness warmup regimes, and proposes choosing $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$8 rather than $F_{\rm drive}(h_0)=-\frac{V_{\rm bulge}}{h_0}$9 so as to exploit the loss catapult mechanism more directly (Kalra et al., 2024).

“Feature Learning Beyond the Edge of Stability” connects beyond-EOS catapult behavior to the minibatch-loss Taylor coefficients and hidden-feature geometry. In its language, the catapult phase is the transient regime beyond the EOS threshold where the loss oscillates, sharpness decreases, hidden features become more orthogonal, and training later stabilizes. The paper uses

$$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$0

for sharpness and $$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$1 as an NTK proxy, and studies a homogeneous MLP with widths $$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$2 under the Normalized Update Parameterization $$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$3. Empirically, larger $$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$4 permits larger stable learning rates, starts the catapult phase earlier, makes it last longer, yields flatter solutions, and produces more orthogonal hidden representations (Terjék, 18 Feb 2025).

A separate line of work ties catapult to objective-function structure. “Good regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult” treats catapult as the pre-EoS or de-sharpening stage in which loss may spike while GD escapes a sharp region. It argues that such behavior is more likely for objective functions with good regularity and may disappear under bad regularity, producing one-sided stability instead. The same paper frames balancing, edge of stability, and catapult as related manifestations of a broader large-learning-rate mechanism (Wang et al., 2023). “Implicit bias of deep linear networks in the large learning rate phase” similarly emphasizes that, for logistic loss, the catapult picture depends on whether data are separable or non-separable; under a degenerate non-separable setting the paper proves a catapult phase with decreasing NTK and flatter convergence, while for separable data the finite-minimum comparison is no longer the right framing (Huang et al., 2020).

6. Biological and engineering analogues

Outside fission, wakes, and optimization, the term “catapult” often refers to a release mechanism, and a “catapult phase” is then the short release interval. In basidiomycete fungi, spores are launched by a surface tension catapult: Buller’s drop coalesces with an adaxial drop, surface area decreases, and the released surface energy is converted into kinetic energy of the spore–drop complex. The ejection velocity is modeled as

$$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$5

with maximum launch speed at

$$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$6

The paper’s central conclusion, however, is not that fungi maximize speed: the data suggest Buller’s drop is regulated for maximum packing efficiency, with a packing-favorable regime around $$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$7 rather than the maximum-speed optimum (Iapichino et al., 2019).

In the spotted lanternfly, the adult mechanism is explicitly phased as droplet growth, spring loading, and rapid catapult release. Adults use a longer stylus with a compliant resilin basal region, maintain stylus–droplet contact through a finite compression phase, and release droplets with both translational and rotational momentum. The reduced-order adult model is a coupled two-spring system,

$$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$8

with the droplet as an upper capillary spring and the stylus plus elastic base as a lower oscillator. The adult ejection speed is

$$\dot{Q}_{\rm bulge}=m\rho \bar v \int U(s)^2\,d^2\sigma =K(h_0)\dot h_0^2, \qquad F_{\rm frict}=-K(h_0)\dot h_0.$$9

the stylus tip speed is

$\dot h_0(t=0)\approx -0.2\,v_F,$0

and the speed ratio is

$\dot h_0(t=0)\approx -0.2\,v_F,$1

Nymphs, by contrast, use capillary rectification rather than elastic catapulting, so ontogeny changes how stylus motion is coupled to the droplet at release (Ha et al., 14 May 2026).

The walking-robot literature uses the catapult analogy for elastic energy storage and release in push-off. In EcoWalker-2, passive knee flexion initiation delays the Start of Ankle Plantar Flexion by about $\dot h_0(t=0)\approx -0.2\,v_F,$2 of the gait cycle relative to active initiation, leading to an $\dot h_0(t=0)\approx -0.2\,v_F,$3 larger increase in trailing-leg horizontal momentum and a $\dot h_0(t=0)\approx -0.2\,v_F,$4 larger magnitude increase in the center-of-mass momentum vector during the step-to-step transition. The paper interprets knee flexion timing as part of the release mechanism: the catapult is not only about stored elastic energy, but also about when it is unloaded relative to leading-leg touch-down (Kiss et al., 2024).

Several other papers use “catapult” without a formal “phase” as a technical stage or metaphor. “Schrodinger’s catapult” describes the active release phase in which pump tones parametrically convert a high-Q cavity state into an output resonator mode,

$\dot h_0(t=0)\approx -0.2\,v_F,$5

allowing release in about $\dot h_0(t=0)\approx -0.2\,v_F,$6 ns, roughly three orders of magnitude faster than the cavity’s intrinsic lifetime, with estimated efficiency greater than $\dot h_0(t=0)\approx -0.2\,v_F,$7 (Pfaff et al., 2016). “Earth–Mars Transfers with Ballistic Capture” does not use the phrase “catapult phase” verbatim, but its distant ballistic-capture segment after the matching maneuver $\dot h_0(t=0)\approx -0.2\,v_F,$8 is the closest analogue: a no-thrust interval in which the spacecraft is carried naturally into temporary Mars capture by stable-set geometry (Topputo et al., 2014). By contrast, “CATAPULT: A CUDA-Accelerated Timestepper for Alpha Particles Using Local Tricubics” uses the term as an acronym—“CUDA-Accelerated Timestepper for Alpha Particles Using Local Tricubics”—and explicitly states that it does not define a special “CATAPULT phase”; the relevant meaning of phase there is simply phase space under guiding-center evolution (Czekanski et al., 8 Apr 2026).

A plausible implication of this wider usage is that the phrase remains most coherent when it names a short release interval rather than an enduring state. Whether the underlying medium is a fission fragment, a shear layer, a training trajectory, a stylus–droplet system, or a microwave cavity, the catapult phase is typically the interval in which previously accumulated structure is converted into directed motion, emission, or stabilization.