Extended Interface PINN (XI-PINN)

• XI-PINN is a physics-informed neural network method that converts low-regularity interface solutions into a smoother, higher-dimensional function using an extended variable.
• The approach employs a level set representation and a single-network formulation to integrate PDE, boundary, initial, and interface jump conditions without subdomain partitioning.
• XI-PINN demonstrates improved error control and faster NTK convergence compared to traditional PINN methods, highlighting its potential for moving-interface problems.

Searching arXiv for the XI-PINN paper and closely related interface/domain-decomposition PINN literature. Extended Interface Physics-Informed Neural Networks (XI-PINNs) are a class of physics-informed neural network methods for partial differential equations with moving or stationary interfaces, especially when the solution itself, or its derivatives and fluxes, exhibit low regularity across an interface. In the formulation explicitly named XI-PINN, the interface is represented by a level set function and the solution is not approximated directly as $u(\mathbf{x},t)$; instead, a single neural network approximates an extended function $\widetilde{u}(\mathbf{x},t,z)$, where the additional variable $z$ is constructed from interface information. The physical solution is recovered by restriction through $$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$. This use of an extended variable technique is intended to convert a low-regularity interface solution into a smoother function in a higher-dimensional space, while retaining a single-network representation over the full domain (Bi et al., 2 Aug 2025). Within the broader PINN literature, XI-PINN belongs to the family of interface-aware and domain-decomposition-inspired extensions that emerged from conservative PINNs and XPINNs, but it differs from subdomain-wise approaches by encoding interface structure through an augmented input rather than assigning one network to each subdomain (Raissi et al., 2024).

1. Conceptual definition and lineage

XI-PINN is formulated for moving interface problems in which a fixed spatial domain $\Omega \subseteq \mathbb{R}^d$ is partitioned into two time-dependent subdomains $\Omega^+(t)$ and $\Omega^-(t)$ by a smooth moving interface $\Gamma(t)$. The model problem treated in the explicit XI-PINN paper is the parabolic interface equation

$$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$

with interface jump conditions

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$

and piecewise-constant diffusion coefficient

$\widetilde{u}(\mathbf{x},t,z)$0

The interface motion is driven by a velocity field through

$\widetilde{u}(\mathbf{x},t,z)$1

This is the setting in which the XI-PINN method is defined (Bi et al., 2 Aug 2025).

The central conceptual move is the use of an extended variable technique. Rather than approximating $\widetilde{u}(\mathbf{x},t,z)$2 directly, XI-PINN approximates an extended function $\widetilde{u}(\mathbf{x},t,z)$3, where $\widetilde{u}(\mathbf{x},t,z)$4 is derived from the interface. The solution is reconstructed by

$\widetilde{u}(\mathbf{x},t,z)$5

The paper states that this allows the discontinuous or low-regularity interface solution to be represented as the restriction of a smoother function in a higher-dimensional space (Bi et al., 2 Aug 2025).

Within the taxonomy of PINN extensions, XI-PINN sits closest to the “Domain Decomposition” branch described in the PINN review literature. That branch includes cPINN, where flux continuity and state continuity are imposed at interfaces for conservation laws, and XPINN, where arbitrary PDEs are handled by decomposing the domain and enforcing interface residual continuity (Raissi et al., 2024). XI-PINN differs from these methods in architecture: it uses a single neural network across the full domain, rather than one PINN per subdomain. A plausible implication is that XI-PINN can be interpreted as an interface-centered alternative to multi-network domain decomposition when the main difficulty is low regularity induced by a moving interface rather than a need for independent subdomain-specific subnetworks.

2. Level set representation and extended-variable construction

The moving interface is represented by a level set function $\widetilde{u}(\mathbf{x},t,z)$6, with sign structure

$\widetilde{u}(\mathbf{x},t,z)$7

Hence,

$\widetilde{u}(\mathbf{x},t,z)$8

The method also uses an indicator function

$\widetilde{u}(\mathbf{x},t,z)$9

These quantities define the extension variable $z$0 used by the neural network (Bi et al., 2 Aug 2025).

The paper distinguishes two cases. If the interface value jump is nonzero, $z$1, then

$z$2

If the interface value jump vanishes, $z$3, then

$z$4

The paper explicitly notes that when $z$5, using $z$6 can “slight improve accuracy since the neural network function ... enforces continuity conditions on the whole domain $z$7” (Bi et al., 2 Aug 2025). In encyclopedic terms, this means the choice of extended coordinate is aligned with the regularity class of the interface solution: side-label information is used for discontinuous solutions, while distance-to-interface information is used for continuous solutions with derivative or flux jumps.

The level set function may be known explicitly or constructed indirectly. When the velocity field extends to the whole domain, $z$8 may satisfy the transport equation

$z$9

When no explicit level set is available, the paper proposes learning an inverse flow map $$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$0 and defining

$$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$1

The neural approximation is

$$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$2

Thus, XI-PINN does not require the interface to be analytically prescribed, but it does require a geometric representation, either directly or through a learned inverse map (Bi et al., 2 Aug 2025).

This level-set-centered treatment links XI-PINN to the broader moving-interface PINN literature. A level set PINN for transport problems similarly represents the interface as the zero contour of $$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$3 and evolves it through

$$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$4

but that work does not introduce the extended variable technique or interface jump enforcement and remains a global level set transport method rather than an interface PDE solver in the XI-PINN sense (Mullins et al., 4 Feb 2025). This suggests that XI-PINN uses the level set not merely as a tracking device but as a structural input to the solution ansatz itself.

3. Neural representation, residual construction, and interface losses

XI-PINN uses one DNN $$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$5 with input dimension $$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$6: spatial coordinates, time, and the extension variable $$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$7. The physical solution is evaluated as

$$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$8

The optimization problem is written as

$$\widetilde{u}(\mathbf{x},t,z(\mathbf{x},t))=u(\mathbf{x},t)$$9

with five residual blocks: PDE, boundary, initial, interface flux, and interface value jump (Bi et al., 2 Aug 2025).

The PDE residual term is

$\Omega \subseteq \mathbb{R}^d$0

evaluated on $\Omega \subseteq \mathbb{R}^d$1. The boundary and initial terms are

$\Omega \subseteq \mathbb{R}^d$2

and

$\Omega \subseteq \mathbb{R}^d$3

Interface conditions are enforced through separate losses: $\Omega \subseteq \mathbb{R}^d$4 and

$\Omega \subseteq \mathbb{R}^d$5

These losses are evaluated at interface-time samples on $\Omega \subseteq \mathbb{R}^d$6 (Bi et al., 2 Aug 2025).

A distinctive feature of XI-PINN is that the PDE operator is evaluated through the chain rule because $\Omega \subseteq \mathbb{R}^d$7 depends on $\Omega \subseteq \mathbb{R}^d$8. The paper gives

$\Omega \subseteq \mathbb{R}^d$9

$\Omega^+(t)$0

and

$\Omega^+(t)$1

This means the interface geometry enters the PDE residual through $\Omega^+(t)$2, $\Omega^+(t)$3, $\Omega^+(t)$4, and $\Omega^+(t)$5, rather than only through separate interface penalties (Bi et al., 2 Aug 2025). This is the defining analytical distinction between XI-PINN and standard interface PINNs that merely add jump losses to an otherwise ordinary PINN ansatz.

4. Relation to XPINN, cPINN, and neighboring interface PINN methods

The review “Physics-Informed Neural Networks and Extensions” places interface- and decomposition-based PINNs under “Domain Decomposition,” describing cPINN as a method for conservation laws with “flux continuity at the interfaces together with state continuity” imposed in the loss function, and XPINN as the extension to arbitrary PDEs where “the continuity of the residuals was enforced along the interfaces” (Raissi et al., 2024). In this lineage, cPINN uses one PINN per subdomain and emphasizes conservative coupling; XPINN uses one PINN per subdomain and generalizes to arbitrary PDEs and space-time decomposition. XI-PINN inherits the same interface-aware motivation but replaces subdomain-wise decomposition by an extended-variable single-network architecture (Bi et al., 2 Aug 2025).

The distinction is technically significant. XPINN assembles a piecewise global solution from subdomain networks and couples them through interface continuity, residual continuity, and optionally flux continuity (Shukla et al., 2021). XI-PINN instead keeps a single solution network over the full domain and inserts interface structure through the augmented coordinate $\Omega^+(t)$6. This suggests that XI-PINN addresses interface-induced low regularity by modifying the representation space, whereas XPINN addresses scalability and localization by modifying the domain partition.

The difference becomes especially sharp when compared with interface PINNs that explicitly solve separate subnetworks in $\Omega^+(t)$7 and $\Omega^+(t)$8. For elliptic interface problems, a dual-network KAN or MLP architecture has been used in which

$\Omega^+(t)$9

with interface losses for

$\Omega^-(t)$0

That framework uses one network per subdomain and explicit penalty terms on interface jumps (Xin et al., 2 Feb 2026). XI-PINN solves a conceptually similar class of interface conditions, but its architecture is not dual-network or multi-network; it is extended-input and single-network (Bi et al., 2 Aug 2025).

For moving discontinuities in hyperbolic PDEs, XPINN has also been used in an interface-centered manner by splitting the domain into evolving pre-shock and post-shock regions and coupling subnetworks through the Rankine–Hugoniot jump condition rather than continuity (Rehman et al., 5 Nov 2025). This shows that the interface-PINN landscape contains at least three distinct mechanisms: subdomain decomposition with continuity or residual matching, subdomain decomposition with physical jump laws, and extended-variable single-network lifting. XI-PINN belongs to the third category.

A further contrast comes from hard-constrained interface PINNs. In hard-constrained windowing and buffer approaches, interface continuity and flux balance are embedded directly into the ansatz or corrected algebraically rather than enforced by soft penalties. Those methods use subdomain-specific subnetworks and are presented as an alternative to the standard soft-constraint interface PINN formulation (Chung et al., 9 Apr 2026). XI-PINN remains a soft-constraint method: it retains penalty-based losses for interface jumps, but changes the representational geometry of the neural approximation (Bi et al., 2 Aug 2025). This suggests that XI-PINN is orthogonal to the soft-versus-hard constraint distinction: its main novelty lies in extension rather than in the loss-enforcement paradigm.

5. Theoretical analysis: approximation, sampling, interface perturbation, and NTK

The XI-PINN paper gives a formal a priori analysis, developed mainly for the case $\Omega^-(t)$1. The central theorem decomposes the total solution error in $\Omega^-(t)$2 into three terms: $\Omega^-(t)$3 The three components are approximation error, statistical or quadrature error from finite sampling, and optimization error (Bi et al., 2 Aug 2025).

The approximation term is tied to the extended-variable construction: the theorem assumes the existence of an extension $\Omega^-(t)$4. This suggests that the whole analytical argument depends on the central XI-PINN hypothesis that the extended representation is smoother than the original interface solution. The paper uses standard DNN Sobolev approximation results to conclude that for any $\Omega^-(t)$5, one can select $\Omega^-(t)$6 such that

$\Omega^-(t)$7

The implication is not that XI-PINN automatically improves approximation in all settings, but that if the EVT genuinely lifts the interface singularity into a smoother ambient function, then DNN approximation theory becomes more favorable (Bi et al., 2 Aug 2025).

For the discrete-to-continuous loss gap, the paper derives a statistical estimate that decays like $\Omega^-(t)$8, up to logarithmic factors, over the four sample classes $\Omega^-(t)$9, $\Gamma(t)$0, $\Gamma(t)$1, and $\Gamma(t)$2. That bound is written as

$\Gamma(t)$3

This places XI-PINN within the now-standard PINN error taxonomy of approximation, generalization, and optimization (Bi et al., 2 Aug 2025).

The paper also analyzes geometry approximation error. If the level set is approximated by a neural network and the approximate interface $\Gamma(t)$4 differs from the exact interface by Hausdorff distance

$\Gamma(t)$5

then, under $\Gamma(t)$6 and $\Gamma(t)$7,

$\Gamma(t)$8

This gives an explicit perturbation pathway from geometric level-set error to PDE solution error (Bi et al., 2 Aug 2025).

The NTK analysis is used to explain training speed. Following standard NTK dynamics, the paper defines

$\Gamma(t)$9

and interprets larger NTK trace-based metrics as faster convergence proxies (Bi et al., 2 Aug 2025). In the first scalar example, XI-PINN yields

$$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$0

while Vanilla-PINN yields

$$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$1

The paper presents this as numerical evidence that XI-PINN has a more favorable NTK spectrum and faster training convergence than conventional PINNs (Bi et al., 2 Aug 2025). This suggests that the benefit of the extended coordinate is not only approximation-theoretic but also optimization-related.

6. Numerical demonstrations, practical workflow, and scope

The paper evaluates XI-PINN on scalar parabolic moving interface problems in two and three dimensions and on the Oseen equations. The implementation uses fully connected DNNs, typically with 4 layers and 64 hidden units, uniformly sampled training points, the Levenberg–Marquardt algorithm, a maximum of 5000 iterations, and stopping when the loss falls below $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$2. Computations are reported on an RTX 4090D GPU (Bi et al., 2 Aug 2025).

In the first scalar 2D example, the domain is $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$3, the interface is a rotating circle with exact level set

$$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$4

and $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$5, $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$6. For $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$7, $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$8, and training points $$\left\{ \begin{aligned} \partial_t u - \nabla \cdot \left( \beta \nabla u\right) &= f \quad &&\mathrm{in}\; \Omega = \Omega^+(t) \cup \Omega^-(t), \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, t\right) &= g \quad &&\mathrm{on}\; \partial \Omega, \quad t \in [0, T_{\text{end}}], \ u\left( \cdot, 0\right) &= u_0 \quad &&\mathrm{in}\; \Omega = \Omega^+(0) \cup \Omega^-(0), \end{aligned}\right.$$9, the reported errors are

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$0

The paper further reports that increasing network width and depth improves accuracy, and that reducing training samples degrades it in a manner consistent with the statistical theory (Bi et al., 2 Aug 2025).

In the second scalar 3D example on $$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$1, the interface is a rotating and rising ellipsoid with velocity

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$2

again with $$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$3, $$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$4. The total errors are

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$5

This is presented as evidence that the method extends to 3D without an obvious loss of fidelity (Bi et al., 2 Aug 2025).

The Oseen examples are especially important because they demonstrate that XI-PINN is not limited to scalar parabolic diffusion. In the first Oseen case, the method solves a two-dimensional moving-interface problem with a star-shaped initial interface and velocity

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$6

using a learned level set through the inverse flow map network. The inverse-map approximation error is

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$7

and the solution errors are

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$8

A second Oseen example with large deformation uses five neural networks for the inverse map to avoid injectivity failure and reports

$$[u]_{\Gamma(t)} = h_D, \qquad [\beta \nabla u \cdot \mathbf{n}]_{\Gamma(t)} = h_N,$$9

Here the multi-network component is used only for geometry learning, not for the XI-PINN solution representation itself (Bi et al., 2 Aug 2025).

The practical workflow follows a fixed sequence. First, represent the interface by a known level set or learn it through an inverse flow map. Second, sample four point sets: interior space-time points, boundary-time points, initial points, and interface-time points. Third, construct the extension variable $\widetilde{u}(\mathbf{x},t,z)$00 from $\widetilde{u}(\mathbf{x},t,z)$01 or $\widetilde{u}(\mathbf{x},t,z)$02. Fourth, train a single DNN $\widetilde{u}(\mathbf{x},t,z)$03 against the sum of PDE, boundary, initial, and interface losses. If the level set is unknown and deformations are large, adaptively partition time and retrain short-time inverse-map subnetworks when the Jacobian determinant criterion indicates loss of injectivity (Bi et al., 2 Aug 2025).

Several caveats are explicit. The main rigorous error theory is developed for $\widetilde{u}(\mathbf{x},t,z)$04. The method assumes a sufficiently smooth extension $\widetilde{u}(\mathbf{x},t,z)$05. When the level set is unknown, geometry learning becomes an additional stage with its own injectivity constraints. The paper’s NTK-based convergence claim is supported by theory and experiments but is not phrased as a universal theorem that XI-PINN always dominates every vanilla PINN configuration (Bi et al., 2 Aug 2025).

In the broader interface-PINN literature, this positions XI-PINN as a distinct method family: neither a pure level set transport PINN, nor a cPINN/XPINN-style multi-subdomain decomposition, nor a hard-constrained ansatz method. Its defining feature is the use of an interface-derived extension variable to regularize representation while preserving explicit interface residuals and moving-geometry handling (Bi et al., 2 Aug 2025). A plausible implication is that XI-PINN is most naturally suited to moving-interface problems where the dominant obstacle is low regularity across a geometrically evolving interface and where a single-network approximation is preferable to subdomain-wise model partitioning.