HollowFlow: Multidisciplinary Flow Efficiency

• HollowFlow is a multidisciplinary concept characterized by efficient flow manipulation in computational models and physical systems using non-backtracking algorithms and transformation media.
• It achieves significant improvements, including up to 100× speed-ups in ML applications and drag-free hydrodynamic cloaking through anisotropic permeability designs.
• The approach enhances mass transfer in sinusoidal hollow fiber membranes with up to 40% fouling reduction and enables programmable morphogenesis in adaptive membranes.

HollowFlow is a term appearing across diverse research subfields, associated with efficient likelihood computation in normalizing flows using non-backtracking message passing (Gloy et al., 24 Oct 2025), advanced hydrodynamic cloaking via transformation media (Urzhumov et al., 2011), sinusoidal hollow fiber membrane engineering for enhanced mass transfer (Luelf et al., 2018), and hydraulic network-induced morphing of thin membranes (Luo et al., 2022). Its applications span scientific machine learning, fluid mechanics, soft matter physics, and membrane technology. The following sections detail the technical underpinnings and practical implications of HollowFlow in these contexts.

1. HollowFlow in Scientific Machine Learning and Normalizing Flows

In normalizing flow models, sample likelihood evaluation typically requires $n$ backward passes through the model for a system of size $n$, causing prohibitive computational costs in large-scale problems. HollowFlow (Gloy et al., 24 Oct 2025) addresses this by imposing a block-diagonal Jacobian structure via a specialized non-backtracking graph neural network (NoBGNN) and message passing algorithm known as Hollow Message Passing (HoMP).

Key design aspects:

• The NoBGNN restricts message updates such that node representations are not contaminated by their own features, enforced by constructing the line graph $L(G)$ and systematically pruning connections using a time-dependent backtracking array.
• The resulting Jacobian $J_b(x)$ of the learned vector field $b$ decomposes as $J_b(x) = J_c(x) + J_\tau(x)$, where $J_c(x)$ is block-hollow and $J_\tau(x)$ is block-diagonal, leading to a divergence calculation that requires only $d$ backward passes (for node feature dimension $d$), independent of $n$.
• For $k$-nearest neighbor (kNN) graphs, the overall computational cost of one integration step is reduced from $O(T n^3 d)$ (fully connected baseline) to $O(n(T^{(lg)} k^2 + d k))$ for HollowFlow, where $T$ is the number of integration steps per sample.

Experimental validation on Boltzmann Generator tasks with up to 55-dimensional Lennard–Jones systems demonstrates speed-ups up to $10^2\times$ with only minor trade-offs in sample quality. The framework admits generalization to equivariant (e.g., PaiNN, E3NN) and attention-based architectures, providing a systematic method for adapting any permutation equivariant GNN or attention mechanism to operate in HollowFlow mode.

2. Transformation Media and Hydrodynamic Cloaking

HollowFlow is featured as a physical concept in fluid mechanics related to hydrodynamic cloaking (Urzhumov et al., 2011). Inspired by transformation optics, the approach mathematically manipulates the flow of fluids around three-dimensional objects using spatial coordinate transformations:

• A concentric shell of anisotropic, spatially varying permeability material surrounds an impermeable object, “bending” streamlines such that the external flow is undisturbed, achieving "cloaking".
• The required permeability tensor in the shell is dictated by the transformation: $$\kappa = \kappa_r(r) \hat{r}\hat{r} + \kappa_t(r)(\hat{\theta}\hat{\theta} + \hat{\phi}\hat{\phi})$$ with explicit formulas for radial and tangential components, e.g., $$\kappa_r(r) = \kappa_0 \frac{b}{b-a} \left( \frac{r-a}{r} \right)^2$$, $\kappa_t(r) = \kappa_0 \frac{b}{b-a}$.
• For free-space exterior, the shell permeability profile is optimized so the disturbance coefficients $A$ and $B$ vanish in creeping flow, thereby eliminating downstream wake and viscous drag ($B=0$ ensures zero net momentum flux at infinity).
• The necessity of negative permeability regions is interpreted as demanding “active” media, which would locally impart energy, e.g., using embedded micro-pumps.

Practical implications include the design of wake-free propulsion systems and drag-free coatings for moving bodies in fluids, extending the principle to microfluidics and the control of laminar/turbulent flows.

3. Sinusoidal Hollow Fiber Membranes and Mass Transfer Enhancement

Under the HollowFlow designation, (Luelf et al., 2018) introduces a dynamic membrane spinning process producing sinusoidal hollow fibers:

• A pulsation module imposes a periodic fluctuation in bore fluid flow rate during fiber extrusion, inducing rhythmic diameter variations along the fiber axis.
• The geometry engenders secondary flow structures (Bellhouse effect), with alternating narrowing and widening causing localized vortices and enhanced radial mixing.
• Mathematical characterization of mass transfer utilizes a modified Sherwood number: $$Sh = 0.023 Re^{0.8} Sc^{0.33} (1 + \alpha \sin(\omega x))$$, reflecting the amplitude and frequency of sinusoidal lumen modulation.
• Experimental data reports 20–30% higher oxygen transport and substantial fouling mitigation (up to 40% reduction) compared to straight fibers.

This methodology is compatible with existing fiber membrane technology and enables improved performance in gas–liquid contactors, medical dialysis membranes, and filtration applications.

4. Hydraulic Flow Networks and Programmable Membrane Morphogenesis

HollowFlow’s conceptual scope incorporates hydraulic network-actuated shape transformations in thin membranes (Luo et al., 2022):

• Membranes are modeled as spring networks overlaid with an embedded flow network, where each node features a storage capacitor (capacitance $C$), and channels have hydraulic resistance ($R$).
• Fluid redistribution causes local swelling; elastic bonds adjust their rest lengths as a function of local fluid content: $$\rho_{jk}(t) = \rho_0 + \frac{m}{2} \left[ \frac{W_j(t)}{W_{max}} + \frac{W_k(t)}{W_{max}} \right]$$.
• The coupled system evolves via quasi-static mechanical energy minimization, with out-of-plane buckling and curvature analyzed through angular-deficit-derived Gaussian curvature metrics.
• Quantitative analysis reveals a strong positive correlation between local fluid content and membrane curvature, especially in networks with sparse major veins.

Biologically inspired by flower petal blooming, this mechanism underpins reconfigurable morphing materials, soft robotics actuators, and adaptive surfaces.

5. Comparative Analysis and Performance Metrics

A comparative summary of HollowFlow’s instantiations across domains is presented in the table below:

Context/Areal	HollowFlow Principle	Performance Outcome
Scientific ML (Gloy et al., 24 Oct 2025)	Non-backtracking message passing	$\mathcal{O}(n^2)$ speed-up (up to $100\times$ ESS/compute)
Fluid Cloaking (Urzhumov et al., 2011)	Transformation media, anisotropic permeability	Elimination of wake/drag; potential for active propulsion
Fiber Technology (Luelf et al., 2018)	Sinusoidal diameter modulation	25% enhanced oxygen transfer, 40% fouling reduction
Membrane Morphogenesis (Luo et al., 2022)	Hydraulic network-induced swelling	Programmable 3D morphing correlated to fluid content/vein architecture

Each variant leverages HollowFlow to mitigate fundamental bottlenecks: computational scaling (ML), dissipation/wake (hydrodynamics), transport limitation (membrane technology), and shape control (soft matter).

6. Implications and Outlook

The overarching theme underlying HollowFlow across these studies is the targeted manipulation of flow—computational or physical—using tailored architectures or media that enforce a hollow, non-redundant structure. The mathematical principle, whether in the form of block-diagonal Jacobians in neural algorithms or spatial permeability profiles in media, enables order-of-magnitude improvements in efficiency and control.

A plausible implication is that as the methodology for non-backtracking message passing and transformation media matures, HollowFlow-type architectures will become standard in both large-scale probabilistic modeling and experimental setups seeking reduced dissipation and tunable morphogenesis.

Potential future developments include the extension to:

• Long-range interaction models in scientific ML (e.g., Ewald summation augmentation for nonlocal phenomena)
• Active metamaterials and programmable porous media for energy-efficient propulsion and adaptive hydrodynamics
• Multi-headed HollowFlow variants balancing expressivity and computational cost
• Integration of flow-controlled shape morphing in robotics and tissue engineering

HollowFlow thus represents a convergent principle for efficient flow – whether in data, fluids, or morphogenetic mechanics – and its continued evolution is anticipated to impact both computational modeling and physical engineering domains.