Conservative PINNs for Scalable PDEs

• Conservative PINNs (CPINNs) are neural architectures that decompose spatial domains to solve PDEs by explicitly enforcing conservation laws at subdomain interfaces.
• They employ local loss terms for data fidelity, PDE residuals, and flux matching, ensuring consistency and accuracy across non-overlapping subdomains.
• The method supports parallel training and flexible local hyperparameter tuning, making it ideal for forward and inverse multi-scale, multi-physics PDE problems.

Conservative Physics-Informed Neural Networks (CPINNs) are a class of physics-informed machine learning solvers that address partial differential equations (PDEs) by leveraging domain decomposition methods—specifically, in the spatial domain—to enable scalable, parallelizable, and locally adaptable neural network architectures. CPINNs impose conservation laws and interface conditions explicitly at subdomain boundaries, enabling effective handling of complex, multi-region, and multi-scale physical problems within the broader framework of domain-decomposed PINNs. They are closely related to Extended PINNs (XPINNs), which generalize these concepts to both space and time decompositions. CPINNs hold distinct computational and modeling advantages for forward and inverse PDE problems involving heterogeneous physics and geometries.

1. Mathematical Formulation and Domain Decomposition Structure

Let $\Omega\subset\mathbb{R}^d$ denote the physical domain for a given PDE. CPINNs decompose $\Omega$ into $N_{sd}$ non-overlapping subdomains: $$\Omega = \bigcup_{q=1}^{N_{sd}} \Omega_q, \qquad \Omega_p \cap \Omega_q = \emptyset\ \text{for}\ p\neq q.$$ In each subdomain $\Omega_q$, an independent neural network parameterized by $\theta_q$ approximates the target field $u(z)$: $u_q(z) \approx u(z), \quad z \in \Omega_q.$ The global solution is composed as

$$u(z) = \sum_{q=1}^{N_{sd}} u_q(z)\,\mathbb{I}_{\Omega_q}(z),$$

with the piecewise indicator $\mathbb{I}_{\Omega_q}(z)$ defined as 1 inside $\Omega_q$ off interfaces, $1/S$ on $S$-way interfaces, and 0 otherwise. Subdomain boundaries (interfaces) $\Gamma_{qp}$ are central for enforcing physical conservation and consistency.

2. Loss Functional, Interface Conditions, and Conservation Enforcement

CPINNs formulate the overall objective as a sum of local losses over data, residual, and interface collocation points: $$\mathcal{J}_{\mathrm{CPINN}} = \sum_{q=1}^{N_{sd}} \mathcal{J}_q(\theta_q)$$ where each local contribution is

$$\mathcal{J}_q(\theta_q) = W_{u_q}\,\mathrm{MSE}_{u_q} + W_{F_q}\,\mathrm{MSE}_{F_q} + W_{I_q}\,\mathrm{MSE}_{u_\mathrm{avg},q} + W_{I_{F_q}}\,\mathrm{MSE}_{\mathcal{R},q}.$$

The components are defined as:

• Data misfit:

$$\mathrm{MSE}_{u_q} = \frac{1}{N_{u_q}}\sum_{i=1}^{N_{u_q}} |u^{(i)} - u_q(z_u^{(i)})|^2$$

• PDE residual:

$$\mathrm{MSE}_{F_q} = \frac{1}{N_{F_q}} \sum_{i=1}^{N_{F_q}} |\mathcal{L}[u_q](z_F^{(i)}) - f(z_F^{(i)})|^2$$

• Solution/interface continuity (with neighboring subdomains $p\in\mathcal{N}(q)$):

$$\mathrm{MSE}_{u_\mathrm{avg},q} = \sum_{p\in\mathcal{N}(q)} \frac{1}{N_{I_q}} \sum_{i=1}^{N_{I_q}} |u_q(z_I^{(i)}) - \{\!\{ u \}\!\}(z_I^{(i)})|^2$$

• Residual (flux) conservation at interfaces:

$$\mathrm{MSE}_{\mathcal{R}, q} = \sum_{p\in\mathcal{N}(q)} \frac{1}{N_{I_q}} \sum_{i=1}^{N_{I_q}} |\mathcal{L}[u_q](z_I^{(i)}) - \mathcal{L}[u_p](z_I^{(i)})|^2$$

where the average $\{\!\{u\}\!\} = (u_q + u_p)/2$ is used for symmetric interface enforcement.

Conservation is explicitly handled via the residual (flux) matching penalty at interfaces, which ensures that physical fluxes or quantities conserved by the PDE are continuous, thereby satisfying the physical law weakly across subdomain boundaries (Shukla et al., 2021).

3. Parallelization and Computational Strategy

CPINNs are specifically constructed to enable parallel/distributed training. The computational domain is partitioned across processing units (e.g., MPI ranks over CPUs/GPUs), with each subdomain assigned to a local network and optimizer. The essential parallelism is as follows:

• Each rank trains its subnetwork locally using data, residual, and interface points within its subdomain.
• Only interface values (function values and residuals) need to be communicated between neighboring subdomains via non-blocking MPI sends/receives.
• No global synchronization (e.g., gradient averaging) is needed—only point-to-point interface-local exchanges per epoch. This architecture enables effective parallel scaling, minimal communication overhead (as compared to time-space decompositions), and allows fully local hyperparameter tuning and model refinement per subdomain (Shukla et al., 2021).

4. Subdomain-Specific Architectures, Hyperparameters, and Adaptation

Each subdomain $\Omega_q$ employs its own fully connected feed-forward neural network $\mathcal{N}_q^L(z; \theta_q)$, where depth, width, activation functions (e.g., tanh, sin, cos, adaptive activations), and all training hyperparameters (weights for loss terms, learning rates, etc.) can be independently set. Sampling densities (counts of data, residual, and interface points) are also chosen per subdomain.

This flexibility supports local adaptation: subdomains featuring stiff gradients, rapid oscillations, or multi-physics interfaces can be assigned deeper/wider networks and denser collocation, while simpler regions can be modeled more efficiently. No global weight synchronization is performed, supporting highly heterogeneous model architectures and adaptive learning across the domain (Shukla et al., 2021).

5. Advantages, Limitations, and Empirical Performance

Advantages:

• Parallelization: CPINN domain decomposition enables strong and weak scaling across distributed computational resources, as only interface data are exchanged.
• Representation capacity: By subdividing complex domains and enabling local architecture adaptation, CPINNs can effectively approximate multi-scale and multi-physics PDE solutions.
• Hyperparameter flexibility: Subdomain-local tuning enables region-specific model improvement.
• Efficiency: For space-only decompositions, CPINNs exhibit reduced communication costs compared to time-space decompositions (as in XPINNs) (Shukla et al., 2021).

Limitations:

• For problems where temporal decomposition is needed (e.g., transient/multiphysics problems with complex time dependencies), pure CPINN approaches do not apply; XPINNs generalize the domain decomposition to handle both space and time.
• Optimal performance depends on suitable domain partitioning and appropriate interface condition enforcement; poor decomposition can hinder generalization (Shukla et al., 2021).

Empirical Results:

Application to an inverse diffusion problem on a partitioned US map (10 irregular regions) demonstrated that CPINNs reconstruct unknown fields (e.g., conductivity $K(x,y)$ and temperature $T(x,y)$) with pointwise error $\lesssim 10^{-3}$, and exhibit close-to-linear scaling: a $\sim9\times$ speedup from 1 to 10 CPUs and $7$–$9\times$ on V100 GPUs (Shukla et al., 2021).

6. Connections to XPINNs and Related Extensions

CPINNs can be viewed as the spatial specialization of XPINNs, which allow domain decomposition in both space and time. While CPINNs are efficient for spatial-only problems with significant heterogeneity but modest temporal complexity, XPINNs further permit partitioning in complex (possibly non-rectangular) space-time subdomains and more naturally accommodate moving fronts, shocks, or nonstationary dynamics. Extended frameworks such as Augmented PINNs (APINNs) replace hard decomposition and strict interface losses with soft (gating) domain assignment and parameter sharing via a gating network, thereby eliminating explicit interface penalties and potentially improving generalization and stability (Hu et al., 2022).

Empirical and theoretical comparisons indicate that XPINNs and APINNs outperform standard PINNs when local solution complexity (measured via Barron–type norms) is sharply reduced by domain decomposition and the penalty from reduced per-subdomain data points does not dominate. CPINNs, as a class, best match problems for which spatial conservation enforcement is key and communication cost constraints favor purely spatial decompositions (Hu et al., 2021, Hu et al., 2022).

7. Applications and Future Directions

CPINNs have been applied to steady and quasi-steady forward and inverse PDE problems with complex spatial domains and multi-scale phenomena, especially where conservation laws and physical fluxes across subdomain interfaces are central. Notable tested applications include inverse heat conduction, conductivity reconstruction on irregular geographical shapes (e.g., US map), and multi-physics engineering scenarios (Shukla et al., 2021).

Future developments may involve integrating CPINN-style conservation with adaptive domain decomposition, hybrid parallel training strategies, and coupling with Bayesian inference to quantify uncertainties. The interplay between strict conservation enforcement and flexible, data-driven interface regularization (via e.g., gating networks or soft aggregation) continues to be an axis of extension in the evolving domain of physics-informed machine learning (Hu et al., 2022, Landgren et al., 28 Sep 2025).