Mass-Conserving Evolution (MaCE) Insights

• Mass-Conserving Evolution (MaCE) is a framework that rigorously enforces conservation laws in dynamical systems through precise mathematical, computational, and algorithmic strategies.
• It employs numerical methods such as ALE finite element schemes and variational, graph-based approaches to maintain invariant mass across complex spatial and temporal domains.
• Applications of MaCE span reaction–diffusion systems, cellular automata, coagulation–fragmentation models, and mass-conserving neural networks, ensuring accurate physical simulations.

Mass-Conserving Evolution (MaCE) refers to a broad set of mathematical, computational, and algorithmic frameworks for describing, simulating, and analyzing dynamical systems in which an extensive conserved quantity—formally, "mass"—is redistributed in space, configurations, network states, or neural representations, subject to strict local or global conservation laws. The MaCE principle has underpinned advances in reaction–diffusion modeling, phase separation, graph-based dynamics, machine learning architectures, cellular automata, and other domains where conservation properties are essential for physical fidelity, regularization, or computational function. The following sections synthesize the core theoretical, numerical, and algorithmic principles of MaCE, as well as the diversity of its methodological realizations and scientific applications.

1. Mathematical Foundations of Mass-Conserving Evolution

MaCE frameworks are grounded in the enforcement of explicit conservation laws at the dynamical, discrete, or algorithmic level. For systems governed by partial differential equations, this takes the form of global or local conservation constraints:

$\frac{d}{dt} M(t) = 0$

$$M(t) = \int_\Omega u(x,t)\,dx \quad \text{or} \quad \sum_i u_i(t)$$

Depending on context, "mass" may refer to the concentration of chemical or biological species (in reaction–diffusion systems), the sum of state variables in cellular automata, the occupancy in graph Laplacians, or storage nodes in dynamical network models.

In the context of reaction–diffusion systems, MaCE is realized by setting up bulk–surface or multicomponent models subject to mass-exchange reactions and enforcing the sum total over space (and sometimes interfaces) to be invariant under all processes, including internal reactions, transport, and boundary conditions. In the continuum, this leads to continuity equations that respect local conservation via divergence form and global conservation via appropriate coupling across subdomains or interfaces (Mackenzie et al., 2019, Frey et al., 14 Dec 2025).

For coagulation–fragmentation models, conservation is reflected in the evolution equation for the distribution $f(t,x)$ of clusters of size $x$, structured so that the gain and loss terms due to merging and splitting of clusters exactly balance in the first moment, yielding $M_1(t) = \int x f(t,x) dx = \text{const}$, barring pathological phenomena such as gelation or shattering (Laurençot, 2019, Laurençot, 2019).

In discrete systems, e.g. graphs or cellular automata, MaCE is implemented by designing update rules (or discretizations of continuum flows) so that the sum over all sites or nodes is preserved at each step. In general, variational, operator-splitting, or gate-constrained updates are formulated to ensure mass-invariant evolution under arbitrary admissible dynamics (Papadopoulos et al., 16 Jul 2025, Budd et al., 2020, Zhang et al., 22 Jun 2026).

2. Numerical and Algorithmic Realizations

Reaction–Diffusion Systems on Evolving Domains

A primary challenge in MaCE frameworks for PDEs is preserving conservation at the discrete, fully-implemented level, especially in complex spatial or moving domains. Finite element arbitrary Lagrangian–Eulerian (ALE) schemes synthesize mesh movement (by evolving the computational domain along with the material), higher-order spatial discretization, and robust enforcement of conservation. Global discrete conservation is established independently of time step and mesh velocity by ensuring that the mass matrices and boundary exchange terms are constructed so that total mass is invariant under all possible grid motions (Mackenzie et al., 2019).

Graph-Based Flows and Semi-Discrete Variational Schemes

For graph-based MaCE, including Allen–Cahn and Merriman–Bence–Osher (MBO) diffusive flows, conservation is built into both the continuous and time-discrete flows via Lagrange multipliers or variational constraints. The semi-discrete scheme minimizes distance to the diffused previous state under the mass constraint, ensuring that each step is strictly conservative. In the MBO limit, thresholding is performed to maintain exactly the prescribed mass (Budd et al., 2020).

Cellular Automata and Neural Cellular Automata

In CA and NCA, MaCE prescribes mass redistribution updates based on "attractiveness" affinities: mass is moved between neighboring cells so that (a) the total mass is preserved to machine precision, and (b) emergent mesoscale structures cannot collapse or diverge due to unbalanced birth/death dynamics. In MaCE-CA, redistribution weights are normalized exponentials of neighbor affinities; for NCA, local softmax "donate–collect" rules accomplish the same effect after a neural update, even in the presence of learned, nonlinear, or convolutional rules (Papadopoulos et al., 16 Jul 2025, Zhang et al., 22 Jun 2026).

Mass-Conserving Neural Networks

For machine learning, MaCE is realized by explicit mass-balance-constrained architectures, such as the Mass-Conserving Perceptron (MCP), which parameterizes all recurrent or gating functions so that net storage, outflow, and unmodeled loss sum exactly to inflow plus prior storage. Softmax normalization ensures that all gating partitions are convex and sum to unity, imposing strict conservation at each node and timestep, regardless of depth or learned nonlinearities. This bi-directionally connects classical "bucket" models of hydrology and modern gated RNNs (Wang et al., 2023).

3. Application Domains and Model Classes

Reaction–Diffusion and Pattern-Formation in Biology and Physics

Mass-conserving reaction–diffusion (McRD) systems underpin modern quantitative models of protein patterning in cells, particularly bulk-surface switching systems where biophysical activation happens via conformational cycling tied to membrane–cytosol exchange. The Min protein system of E. coli serves as a paradigmatic example, with MaCE-based models explaining oscillatory and spatially heterogeneous protein distributions, pole-to-pole dynamics, and coarsening behavior. The geometric phase-space formalism provides predictive power by mapping reactive equilibria onto nullclines and analyzing pattern selection via mesoscale mass-redistribution potentials (Frey et al., 14 Dec 2025).

MaCE is also foundational in modeling solid–solid phase transitions, where order parameter dynamics are controlled by flux-limited diffusion and strict mass constraints. This allows accurate tracking of sharp interface motion, nucleation vs. spinodal decompositions, and the bifurcation structure as a function of mass and domain length (Burns et al., 2011).

Coagulation–Fragmentation and Population Balance

The theory of mass-conserving evolution underpins classical and contemporary solutions to the coagulation–fragmentation equations with homogeneous or "balanced-growth" kernels. For suitable parameter regimes (e.g., small initial mass), one can establish the global-in-time existence and uniqueness of mass-conserving weak solutions and the emergence of self-similar, scale-invariant dynamics (Laurençot, 2019, Laurençot, 2019).

Cascades and Anomalous Transport

MaCE arises in the modeling of reservoir cascades and generalized transport, replacing classical exponential waiting kernels with q-exponential forms. Here, mass-conserving convolutions yield output distributions governed by stable Lévy laws, resulting in non-Gaussian, anomalously-dispersive hydrographs and revealing that heavy-tailed dispersion emerges from conservation alone, without recourse to stochastic or fractional modeling (Lima et al., 19 Jun 2026).

Machine Learning and Physical Modeling

MaCE principles in neural architectures yield interpretable and physically-faithful ML systems, such as the MCP, which ensure that predictions respect mass-balance at every recurrent step, increasing transparency in geophysical modeling and facilitating coupling with energy or information flows (Wang et al., 2023). In NCA-based reservoir computing, MaCE operates as an inductive bias for self-organized criticality, enhancing the frequency and quality of critical avalanche regimes relevant for information processing, and regularizing the search landscape (Zhang et al., 22 Jun 2026).

4. Structural and Algorithmic Properties

Conservation Proofs and Stability

Mass conservation in these frameworks is generally established via explicit calculation: for each update or flow, the total mass before and after a step can be shown to be equal by summing over all degrees of freedom, using particular properties of the update rules (such as normalization of weights, vanishing boundary flux terms, or convex minimization under the mass constraint) (Mackenzie et al., 2019, Papadopoulos et al., 16 Jul 2025, Budd et al., 2020).

Numerical schemes are engineered for unconditional or broad stability. For instance, the MaCE update in cellular automata maintains stability by adjusting the time step in relation to the spatial coupling; for fully discrete finite element schemes, conservation is independent of mesh velocity or time step size (Mackenzie et al., 2019, Papadopoulos et al., 16 Jul 2025).

Expressivity and Model Class

While MaCE imposes hard constraints, thereby reducing the admissible functional class compared to unconstrained dynamics (e.g., in GRNNs or LSTMs), the inductive bias can increase pattern diversity (soliton abundance in Lenia-CA), encourage richer critical phenomena (SOC in NCA), and ensure physical realism (hydrological runoff, phase separation). In structured graph or PDE settings, the MaCE principle does not impede convergence to expected energies or Lyapunov functionals, allowing robust theoretical guarantees (Budd et al., 2020, Wang et al., 2023).

Extensions and Multiphysics Generalizations

MaCE principles generalize to network flows of other conserved quantities (energy, momentum), with the possibility to learn multiple conservation laws within a single architecture. Coupling mass-conserving subgraphs with energy-conserving or information-flow units leads to compositional, interpretable modeling frameworks for complex, multiphysical and multiscale systems (Wang et al., 2023).

5. Representative Examples and Empirical Results

Domain	Mass-Conserving Quantity	Methodology	Notable Phenomena/Results
Intracellular signaling	Protein copy number	ALE FEM, McRD equations	Quantitative reproduction of Min oscillations; robust pattern selection
Cellular automata (Lenia/NCA)	Lattice mass / visible state sum	MaCE affinity updates	Soliton abundance, intrinsic resource-limited evolution, SOC regularity
Coagulation–fragmentation	Particle size distribution sum	Weak solutions, scaling	Existence of self-similar mass-conserving solutions
Hydrology models (MCP)	Storage, discharge, losses	Mass-conserving RNN	Physically-constrained generalization to complex catchment dynamics
Graph diffusion (Allen–Cahn)	Node occupancy	Variational MBO flows	Maintenance of target mass under phase separation and segmentation

Tested across these domains, MaCE-based models achieve high-fidelity reproduction of empirical patterns (cellular, hydrological), stability and robustness in numerics (second-order convergence, Lyapunov monotonicity), and superiority or parity in learning efficiency and generalization compared to unconstrained algorithms (Mackenzie et al., 2019, Papadopoulos et al., 16 Jul 2025, Wang et al., 2023, Frey et al., 14 Dec 2025, Lima et al., 19 Jun 2026, Zhang et al., 22 Jun 2026).

6. Implications and Future Directions

The MaCE paradigm supports a unified theory for mass-conserving evolution across physical, biological, computational, and algorithmic systems. Key implications include:

• The possibility of replacing ad hoc or phenomenological "open" system models with physically-justified, mass-conserving kernels that admit both analytic tractability and empirical accuracy,
• The use of conservation-imposed inductive biases in learning and search (e.g., evolutionary or gradient-based optimization), which yield richer critical behavior, better parameter regularization, and more interpretable internal dynamics in both synthetic and data-driven settings,
• Extensions toward multiconservation principles (e.g., energy, information), networked or nonlocal conservation, and open-ended evolution within CA frameworks,
• New connections between mesoscale geometrical laws (curvature-driven motion, von Neumann laws) and conservation-saturated nonlinear PDEs.

A plausible implication is that the MaCE principle serves as a minimal yet highly effective mechanism for the emergence of self-organization, criticality, and robust information processing in a broad array of natural and artificial systems, and will continue to inform both model construction and theoretical analysis across disciplines (Papadopoulos et al., 16 Jul 2025, Frey et al., 14 Dec 2025, Wang et al., 2023, Zhang et al., 22 Jun 2026).