Invariant Domain Preservation

• Invariant domain preservation is the principle that updates, maps, or projections keep states within a defined, admissible set to maintain inherent structural properties.
• It spans various fields—from hyperbolic conservation laws to stochastic dynamics and cross-domain synchronization—by enforcing convexity, state-transition rules, and invariant eigenfunctions.
• Practical techniques include convex limiting in finite-element methods, invariant eigenfunction preservation in Koopman approximations, and adaptive domain mappings in data-driven harmonization.

Invariant domain preservation denotes the requirement that an update, projection, synchronization rule, or representation map remain inside a prescribed admissible set while retaining the structural information that defines that set. Across recent literature, the phrase is used in several non-equivalent but mathematically related senses: preserving convex admissible states for hyperbolic conservation laws and stochastic differential equations; preserving regulatory state transitions across multiple synchronized domains; preserving invariant sets in Koopman-based approximations; and preserving task-relevant, causal, or anatomical content under domain adaptation and harmonization (Harmon et al., 24 Jul 2025, Erdogan et al., 17 Mar 2025, Kim, 4 Apr 2026, Seenivasaharagavan et al., 2023, Lan et al., 2024). In each case, the central question is not only whether a map is accurate, but whether it respects the domain on which the underlying model or semantics remains valid.

1. Formal scope and recurring mathematical structure

A standard numerical formulation appears for hyperbolic conservation laws

$$\partial_t \boldsymbol{u} + \nabla \cdot \boldsymbol{f}(\boldsymbol{u}) = \boldsymbol{0},$$

with a convex admissible set

$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$

and an operator $T$ is called invariant-domain-preserving if $$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$, together with conservation of element-wise and global integrals. In recent finite-element AMR work, the admissible sets include $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$ and $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$ (Harmon et al., 24 Jul 2025). For one-dimensional PAMPA schemes, the same idea is expressed through an invariant domain $G\subset\mathbb{R}^m$, with discrete IDP inherited by SSP time integrators because SSP methods are convex combinations of forward Euler steps and $G$ is convex (Abgrall et al., 2024).

A second formulation arises for bounded stochastic dynamics. For SDEs on the open hypercube

$D := (L_1,R_1)\times \dots \times (L_d,R_d),$

the exact solution remains in $D$ when the drift points inward at the boundary and the diffusion vanishes on $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$0. Numerical invariant domain preservation is then built by explicit updates whose componentwise flows preserve left and right bounds and are combined by convex weights (Erdogan et al., 17 Mar 2025).

A third formulation is categorical in spirit. In cross-domain state synchronization, each domain is an abstract deterministic partial state machine $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$1, and preservation is encoded by structure-preserving maps satisfying naturality: $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$2 Here the invariant object is not a positivity set but the regulatory state-transition structure itself (Kim, 4 Apr 2026).

In Koopman-based model reduction, invariant domain preservation appears through $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$3 eigenfunctions. If $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$4, then the level sets

$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$5

are forward-invariant, so preserving these eigenfunctions preserves invariant domains of the dynamics (Seenivasaharagavan et al., 2023).

2. Cross-domain state synchronization and mechanized safety–liveness proofs

A fully formal treatment is given for cross-domain state preservation under Byzantine faults. The development models a finite set of domains $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$6, each instantiating the locale state_machine with finite state set $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$7, finite action set $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$8, deterministic partial transition function $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$9, and terminal set $T$0. The assumptions include terminal_absorbing, transition_closed, and transition_domain. Unidirectional preservation is formalized by state_preservation, bidirectional roundtrip by symmetric_state_preservation, and $T$1-domain consistency by multi_domain_preservation (Kim, 4 Apr 2026).

The core safety theorems are single-step and sequential. Sequential Preservation states that if

$T$2

then

$T$3

Roundtrip guarantees are encoded as

$T$4

from which injectivity of state_map is derived. At the multi-domain level, Cross-Domain Consistency proves that if sync_all succeeds and transition s action = Some s', then every connected domain stores Some s' for that asset, while Sync Isolation proves that synchronization on asset aid leaves every aid' \neq aid unchanged (Kim, 4 Apr 2026).

The regulatory instance uses

$T$5

$T$6

with reg_transition defining 12 valid transitions. The global validity predicate is

$T$7

Within this instance, CONFISCATED is absorbing, FREEZE propagates $T$8 across all connected chains, and valid_state_preservation proves that successful synchronization preserves both cross-domain consistency and lock discipline (Kim, 4 Apr 2026).

The liveness layer introduces priority_system, deadlock_free_locking, and fair_leader_system under the threshold $T$9, hence $$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$0. The formal results are Deterministic Selection via an injective priority function, Deadlock Freedom via lock expiry under timeout > 0, Starvation Bound under bounded fair-leader occurrence, and Eventual Completion by well-founded induction on pending. The paper’s central strengthening is that liveness discharges the honest-node assumption used by safety: deterministic conflict resolution, deadlock freedom, and starvation freedom guarantee the conditions under which the safety theorems apply, thereby promoting conditional safety to an unconditional guarantee under Byzantine faults (Kim, 4 Apr 2026).

3. Hyperbolic conservation laws, AMR transfer, and high-order limiting

For adaptive finite-element discretizations of hyperbolic systems, invariant-domain preservation is formulated as conservative transfer between hierarchical spaces. A three-stage projection

$$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$1

combines element-wise mass projection, nodal averaging, and mass redistribution with hanging-node constraints. The element-wise limited update

$$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$2

uses symmetric limiter coefficients $$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$3 chosen from local convex constraints so that the result remains in $$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$4. The resulting theorems establish conservative IDP element-wise projection, conservative IDP nodal averaging, and conservative IDP mass redistribution (Harmon et al., 24 Jul 2025).

The method is explicit about physically relevant invariant sets. For compressible Euler,

$$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$5

with

$$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$6

and for shallow water,

$$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$7

The practical significance is illustrated by benchmarks: in a dam-break problem with wetting and drying, a conservative but non-IDP projection causes the time step to collapse to $$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$8, whereas the conservative IDP projection maintains $$\boldsymbol{u}_h^n|_K\in\mathcal{A}\Rightarrow T(\boldsymbol{u}_h^n)|_K\in\mathcal{A}$$9; in a Sedov-like blast with JWL EOS, uniform finest level $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$0 requires $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$1 s on $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$2 cores, while AMR finishes in $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$3 s with an overall speedup factor $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$4 (Harmon et al., 24 Jul 2025).

A distinct one-dimensional line of work studies PAMPA schemes. There, the decisive identity is the cell-average decomposition

$\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$5

The analysis shows that if the midpoint value leaves the invariant domain, then no uniform CFL number ensures updated cell averages remain in $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$6 for all configurations. The remedy is a local scaling limiter that moves midpoint and endpoint values toward the cell average while preserving the decomposition exactly. For cell averages, the IDP proof requires an IDP numerical flux and the CFL bound

$\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$7

For point values, the paper proposes an automatic limiter-free reformulation inspired by Softplus and Clipped ReLU. In the Euler case, one sets

$\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$8

so that any point update in $\mathcal{G}_{\mathrm{Euler}}=\{\rho>0,\;p>0\}$9-variables maps back into the positivity set (Abgrall et al., 2024).

A more general 2026 framework treats high-order FV and DG discretizations under explicit and implicit time integration. It limits antidiffusive fluxes around a low-order solution known to preserve all invariant domains and defines coefficients so that the limited solution is a convex combination of invariant-domain-preserving quantities. The construction generalizes flux-corrected transport to systems of conservation laws, supports explicit or implicit Runge–Kutta and time DG integration, and can be iterated to refine the limited solution toward the high-order one while preserving invariant domains (Fanizza et al., 8 May 2026).

4. Stochastic differential equations and compressible Navier–Stokes

For SDEs with diagonal diffusion on a bounded hypercube, invariant-domain preservation begins at the continuous level. The exact system

$\mathcal{G}_{\mathrm{SW}}=\{h>0\}$0

has a unique strong solution satisfying

$\mathcal{G}_{\mathrm{SW}}=\{h>0\}$1

when $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$2, $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$3, and $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$4. The proof uses the Lyapunov function

$\mathcal{G}_{\mathrm{SW}}=\{h>0\}$5

The numerical construction then builds two positivity-preserving flows per component, one guaranteeing $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$6 and one guaranteeing $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$7, and combines them by a convex rule or an indicator-based selection. No projection, reflection, truncation, or taming is required. The Euler version has proven strong mean-square order $\mathcal{G}_{\mathrm{SW}}=\{h>0\}$8,

$\mathcal{G}_{\mathrm{SW}}=\{h>0\}$9

while numerical experiments report that in some cases first-order convergence is observed in practice (Erdogan et al., 17 Mar 2025).

For the compressible Navier–Stokes equations, invariant-domain preservation is achieved by splitting the dynamics into a hyperbolic and a parabolic part. The admissible set is

$G\subset\mathbb{R}^m$0

and for the hyperbolic substep one refines this to

$G\subset\mathbb{R}^m$1

whereas for the parabolic substep one uses

$G\subset\mathbb{R}^m$2

The fully discrete method is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving under the standard hyperbolic CFL condition $G\subset\mathbb{R}^m$3 (Guermond et al., 2020).

The hyperbolic half-steps are explicit and use graph viscosity, auxiliary bar states, and convex limiting to preserve density positivity and an entropy lower bound. The parabolic full step freezes density, advances velocity by Crank–Nicolson, and updates internal energy by a backward-Euler low-order solve together with a Crank–Nicolson high-order correction limited by FCT. The discrete minimum principle gives

$G\subset\mathbb{R}^m$4

and the total-energy update

$G\subset\mathbb{R}^m$5

satisfies a discrete source-work identity. The key distinction from explicit positivity-preserving Navier–Stokes schemes is that no parabolic restriction $G\subset\mathbb{R}^m$6 is imposed (Guermond et al., 2020).

5. Koopman approximations and invariant sets in data-driven dynamics

In Dynamic Mode Decomposition and EDMD, the central preserved property is usually linearity of the Koopman operator. “Invariant Consistent Dynamic Mode Decomposition” extends this by preserving functional relationships among observables and consistency along geometric invariants. For a deterministic autonomous system $G\subset\mathbb{R}^m$7, Koopman eigenfunctions satisfy

$G\subset\mathbb{R}^m$8

When $G\subset\mathbb{R}^m$9, $G$0 is invariant along trajectories, and the level sets

$G$1

are forward-invariant (Seenivasaharagavan et al., 2023).

The constrained regression imposes

$G$2

together with linearized functional constraints such as

$G$3

Under compatibility and rank assumptions, the feasible set admits the explicit parametrization

$G$4

so the constrained least-squares problem reduces to a smaller problem in the nullspaces. The computational overhead is described as minimal and primarily due to the linearity of constraints (Seenivasaharagavan et al., 2023).

The payoff is an exact or approximate enforcement of $G$5 eigen-relations on selected invariant samples. If $G$6, then

$G$7

and left eigenvectors $G$8 satisfy

$G$9

For $D := (L_1,R_1)\times \dots \times (L_d,R_d),$0, the rows of $D := (L_1,R_1)\times \dots \times (L_d,R_d),$1 approximate invariant observables whose level sets delineate invariant domains. The framework also subsumes Affine DMD, by preserving the constant eigenfunction, and Higher-order DMD, by preserving delay-shift relationships. Numerical evidence is given on one-dimensional multistable systems, a Duffing oscillator, and competing limit cycles, where IC-DMD improves constancy of the induced $D := (L_1,R_1)\times \dots \times (L_d,R_d),$2 eigenfunctions on invariant sets relative to vanilla EDMD (Seenivasaharagavan et al., 2023).

6. Domain-invariant representations, harmonization, and factor preservation

In multi-domain neuroimaging harmonization, invariant domain preservation means removing scanner- or site-specific variability while preserving biological and anatomical content. A diffusion-based framework does this by learning a domain-invariant anatomical condition $D := (L_1,R_1)\times \dots \times (L_d,R_d),$3 and combining it with domain embeddings $D := (L_1,R_1)\times \dots \times (L_d,R_d),$4 through AdaIN-modulated layers in a conditional DDPM. The losses are

$D := (L_1,R_1)\times \dots \times (L_d,R_d),$5

with $D := (L_1,R_1)\times \dots \times (L_d,R_d),$6. The model uses a fixed $D := (L_1,R_1)\times \dots \times (L_d,R_d),$7-step schedule, and the proposed “skip sampling” re-noises a target-style sample to $D := (L_1,R_1)\times \dots \times (L_d,R_d),$8 before standard reverse sampling (Lan et al., 2024).

Empirically, the method is evaluated on ADNI 1 and ABIDE II. On ABIDE II, the learned-condition model improves FID relative to GAN baselines in several harmonizations; for Others$D := (L_1,R_1)\times \dots \times (L_d,R_d),$9GU, the reported scores are CycleGAN $D$0, Seg-Renorm $D$1, Ours (edge map) $D$2, and Ours (learned) $D$3. On ADNI 1, downstream PVS count ratio distributions become consistent across all scanners after harmonization to a common scanner style, and t-SNE plots of the domain embeddings collapse from three separated clusters to a single cluster after harmonization (Lan et al., 2024).

In multi-source domain adaptation, “preservation” can mean preserving selected task-relevant domain factors rather than erasing them. Factor-Preserving Domain Adaptation modifies adversarial training by masking the domain loss so that competition is restricted to domains sharing a chosen factor value. On CORe50, the selected factor is “hand side,” identified from PCA on one-to-one transfer matrices. With unsupervised target data, FP-DA reports average accuracy $D$4 and minimum accuracy $D$5, compared with $D$6 and $D$7 for standard adversarial DA with target data, and $D$8 and $D$9 for no DA. The paper’s interpretation is that removing all domain factors can cause negative transfer when a domain factor is simultaneously task-informative or visually close to the object (Schrom et al., 2020).

These practical systems share a direct design principle: the invariant component is represented explicitly and is not inferred solely by global domain confusion. In neuroimaging it is the learned anatomical condition $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$00; in FP-DA it is a factor-consistent grouping that switches off adversarial competition across selected domains (Lan et al., 2024, Schrom et al., 2020).

7. Theoretical critiques, causal structure, and preserver problems

Recent theory makes clear that invariant domain preservation in representation learning is not equivalent to indiscriminate domain invariance. “Support and Invertibility in Domain-Invariant Representations” shows that classical discrepancy bounds do not account for information lost in non-invertible representations and that strict domain invariance can be too strong. The paper introduces a support sufficiency divergence

$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$01

and an explicit non-invertibility cost

$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$02

leading to the bound

$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$03

The central claim is that penalizing distance between densities is often wasteful, whereas support coverage and information retention are the relevant control quantities (Johansson et al., 2019).

A causal reformulation reaches a related conclusion. Under the CISA model, invariant structure is the counterfactually invariant part $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$04 of the features, and the correct observable invariance depends on the causal subtype: anti-causal problems support $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$05, confounded-outcome problems support $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$06, and confounded-descendant problems support $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$07. The paper argues that using the wrong invariance notion can contradict the true invariant structure, and that IRM is justified only in confounded-descendant settings, while anti-causal settings require generalized IRM with label-prior reweighting (Wang et al., 2022).

A separate analytical line recasts invariant risk minimization itself. “Invariant Risk Minimization Is A Total Variation Model” proves that the standard IRM surrogate is a TV-$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$08 model on the learning risk as a function of the classifier variable $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$09,

$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$10

and proposes a TV-$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$11 variant whose coarea geometry promotes piecewise-constant dependence of the risk across environments. The paper states that TV-$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$12 has robust performance in denoising and invariant feature preservation, and its synthetic ablations report a larger fraction of total weight assigned to invariant features than TV-$$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$13 (Lai et al., 2024).

In copula theory, the preserver problem is sharper: the copula of a continuous random vector is preserved unchanged under strictly increasing marginal transformations, whereas decreasing transformations generally produce reflected or survival copulas, and direct argument transformations $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$14 preserve 2-increasingness but typically destroy uniform marginals unless $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$15 is the identity. The paper’s formulation is that copula property is preserved uniquely under increasing transformations, which places invariant domain preservation in the older tradition of preserver problems rather than only in numerical analysis or machine learning (1908.10181).

Taken together, these results rule out a common misconception: preservation is not synonymous with maximal invariance. In numerical analysis, the admissible set is usually convex and explicit; in synchronization, the preserved object is a transition structure; in operator learning, it is an invariant set encoded by $$\mathcal{A}\coloneqq\{\boldsymbol{u}\in\mathbb{R}^m:\Psi^l(\boldsymbol{u})>0,\;l=1,\dots,k\},$$16 eigenfunctions; in representation learning, preservation may require support sufficiency, invertibility, causal alignment, or explicit factor retention rather than complete domain erasure (Johansson et al., 2019, Wang et al., 2022, Lai et al., 2024, 1908.10181).