PARCv2: Physics-Aware Neural PDE Solver

• PARCv2 is a physics-aware CNN that combines explicit finite-difference stencils with data-driven integration to solve nonlinear PDEs.
• It employs a differentiator block for computing spatial derivatives and an integrator block using a hybrid time-stepping scheme to ensure stability.
• Benchmark results indicate reduced RMSE and improved shock handling across fluid dynamics, energetic materials, and reactive solids simulations.

PARCv2 is a physics-aware recurrent convolutional neural network (CNN) designed to model spatiotemporal dynamics governed by coupled partial differential equations (PDEs), particularly those with unsteady, advection-dominated, or shock-driven nonlinearities. The approach combines explicit discretizations of physical principles with data-driven neural modeling, allowing for stable, accurate simulations of complex phenomena including fluid flow, energetic materials under shock, and shear band formation in reactive solids. PARCv2 extends the original PARC differentiator–integrator architecture to provide direct and adaptable representations of spatial derivatives and solution updates, unifying classical numerical time-stepping and convolutional operator learning in a two-stage training procedure (Nguyen et al., 2024, Cheng et al., 8 Oct 2025).

1. Core Architecture and Operator Integration

PARCv2 is structured around a differentiator–integrator loop. At each discrete time step $t_k$, the network updates the state variables (such as velocity, pressure, temperature, microstructure) using a tandem of:

• A differentiator block computing temporal derivatives that explicitly models advection, diffusion, and reaction source terms using physics-aware convolutional stencils.
• An integrator block employing a hybrid scheme: a classical low-order time-integrator (e.g., forward Euler, Heun, or fourth-order Runge–Kutta [RK4]) provides the backbone for stability, while a CNN-based corrector learns high-order residuals, elevating effective accuracy.

The general time update reads: $$\mathbf{x}_{k+1} = \mathbf{x}_k + \Psi_x(F_{\mathbf{x}}) + S_x(\mathbf{x}_k, F_{\mathbf{x}})$$ where $F_{\mathbf{x}}$ includes explicit advection ($-(\mathbf{u}\cdot\nabla)\mathbf{x}_k$), diffusion ($k\,\Delta\mathbf{x}_k$), and nonlinear source terms, each represented through convolutional (or upwind finite-difference) stencils (Nguyen et al., 2024, Cheng et al., 8 Oct 2025).

Key architectural enhancements in PARCv2 include:

• Explicit spatial derivatives: Rather than encoding spatial operators indirectly in the CNN filters, derivatives such as $\frac{\partial}{\partial x}$, $\nabla \cdot (\cdot)$, $\Delta(\cdot)$ are hard-wired at initialization as fixed-width convolutions (e.g., central differences, upwind schemes), then relaxed to become learnable—thus enabling adaptation to grid- or data-specific artifacts while preserving the physics-informed structure.
• Advection-reactive systems: The differentiator encompasses full advection–reaction–diffusion equations,

$$\frac{\partial \mathbf{x}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{x} + k\,\Delta \mathbf{x} + \mathbf{R}_x(\mathbf{x},\mathbf{u},\mathbf{c})$$

and, for velocity evolution,

$$\frac{\partial \mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u} + \mathbf{R}_u(\mathbf{x},\mathbf{u},\mathbf{c})$$

with direct input of finite-difference approximations.

• Hybrid integration: The integration step approximates the exact time-evolution integral using both a robust numerical quadrature and a learned corrector, balancing classical solver stability with data-adaptive error control (Nguyen et al., 2024).

In variant applications such as pore collapse in energetic materials, upwind schemes are favored over right-difference stencils to suppress numerical oscillations in weak shocks and to realize robust, conservation-respecting updates (Cheng et al., 8 Oct 2025).

2. Physics-Awareness and Inductive Bias

PARCv2 operationalizes physics-awareness through two principal mechanisms:

• Embedded spatial operators: Convolutional kernels initialized with physically meaningful finite-difference weights act as surrogate derivatives. These are regularized to remain close to their analytic forms, ensuring inductive bias toward the expected behavior of physical fields.
• Modular training separation: The network decomposes the prediction task into learning the instantaneous spatiotemporal derivatives (physics-informed, parameter-efficient), and separately, the integration correction (data-driven, high-order error compensation).

For conservation law systems, the advection term is evaluated either by upwind finite differences (e.g., for advective transport of state variables in compressible systems) or fixed central-difference convolutions, with hard-coded or adaptive boundary conditions (e.g., Neumann padding for shock-driven pore collapse), ensuring the network enforces local conservation within numerical stencil error (Cheng et al., 8 Oct 2025).

No explicit physics-informed loss (such as PDE residual enforcement or divergence penalties) is imposed during training or rollout; instead, conservation and invariance are maintained implicitly by the network’s architectural constraints.

3. Problem Domains, State Variables, and Data Interfaces

PARCv2 is applicable to nonlinear field evolution problems including:

• Fluid dynamics benchmarks: 2D Burgers’ equation and Navier–Stokes flows around obstacles.
• Shock-induced reaction and advection-diffusion systems in energetic materials.
• Shear band formation, pore collapse, and plastic heating in reactive crystalline solids.

The typical state tensor $s^t$ comprises up to five channels (temperature $T$, pressure $p$, microstructure indicator $\mu$, and velocity components $u_x, u_y$), tracked on physical grids of varying size (e.g., $64 \times 64$, $128 \times 256$), and advanced in time with problem-appropriate step sizes (Nguyen et al., 2024, Cheng et al., 8 Oct 2025).

At inference, the user provides the initial or current state; PARCv2 outputs the next time step prediction, enabling auto-regressive simulation of complex spatiotemporal patterns. The model requires no explicit initial/boundary condition modeling beyond the injection of the first frame and the chosen convolutional padding (e.g., constant or Neumann).

4. Training Procedures and Hyperparameters

Training is staged in two phases:

• Stage 1 (Differentiator training): The hybrid integrator is fixed, and parameter learning targets minimizing the difference between predicted versus true one-step updates for all state channels using a mean absolute error loss:

$$\mathcal{L}_{\rm diff} = \sum_k \|\hat{\mathbf{x}}_{k+1} - \hat{\mathbf{x}}_k - \Psi_x(F_{\mathbf{x}}(\hat{\mathbf{x}}_k))\|_1 + \cdots$$

• Stage 2 (Integrator correction training): The differentiator is frozen, and only the correction networks are trained to minimize the integration residual error:

$$\mathcal{L}_{\rm int} = \sum_k \|\hat{\mathbf{x}}_{k+1} - \hat{\mathbf{x}}_k - [\Psi_x + S_x](\mathbf{x}_k, F_{\mathbf{x}}(\mathbf{x}_k))\|_1 + \cdots$$

Optimization employs Adam with decaying learning rates, with hyperparameters such as batch size and epochs tailored to each benchmark. For example, Burgers’ equation is trained with batch size 16, 100–150 epochs and learning rate $10^{-3}$, while energetic material simulation uses smaller batch sizes and longer (stage-wise) training (Nguyen et al., 2024, Cheng et al., 8 Oct 2025).

In the energetic materials context, a curriculum of one-step and three-step prediction stages is adopted, each comprising thousands of epochs. Preprocessing includes min–max normalization of scalar channels and normalization of velocities, with no added regularization or physics loss (Cheng et al., 8 Oct 2025).

The typical parameter count is $\sim 1.6 \times 10^7$ (dominated by the U-Net or CNN correctors).

5. Benchmarking, Accuracy, and Limitations

PARCv2 has been benchmarked against multiple operator-learning and physics-informed models:

Model	Benchmark Domain	RMSE (Selected Metrics)	Noted Strengths/Failures
PARCv2	Burgers’ / Navier–Stokes / Energetics	RMSE$_u$: 0.0129 cm/s (Burgers), RMSE$_u$: 0.0727 m/s (Navier–Stokes), Temp RMSE: 229.5 K, Pressure RMSE: 1.63 GPa (Energetics)	Accurate sharp fronts, stable long-time rollouts, learns unknown sources
FNO	Same	RMSE$_u$: 0.0289 cm/s (Burgers), 0.2147 m/s (Navier–Stokes), Temp RMSE: 248.4 K	Generalizes but blurs shocks, needs large data
PINN / PIFNO	Same	RMSE$_u$: 0.2307 m/s (Navier–Stokes)	Residual enforcement but suffers in advection-dominated regimes
PhyCRNet / Neural ODE	Same	High RMSE on shocks and velocity; ODE often fails	Blurs sharp features/spectral bias, unstable for high nonlinearity

PARCv2 demonstrates:

• RMSEs approximately half that of FNO on velocity fields.
• PDE residuals near the direct numerical simulation (DNS) ground truth.
• Stable prediction rollouts up to 50 time-steps in shock-driven pore collapse; predecessor models typically degrade after ∼20 steps.
• At high strain rates or extreme conditions, RMSEs increase (e.g., energetic material with very strong shocks), but stability is retained whereas operator-learning baselines diverge.
• Physically relevant geometric errors (shear-band localization, hotspot area/temperature) are minimized compared to alternatives (Nguyen et al., 2024, Cheng et al., 8 Oct 2025).

Principal limitations include:

• Minor divergence-free violations in incompressible Navier–Stokes settings owing to lack of explicit divergence penalization.
• Mild underprediction of distribution tails (extreme temperature/pressure) due to reliance on data-driven rather than residual-based loss.
• All deep-learning baselines, including PARCv2, exhibit spectral bias—weak, high-frequency features (e.g., secondary shear bands) are diffused out during extrapolation.

In terms of computational cost, PARCv2 achieves forward-inference times three to four orders of magnitude faster than DNS on comparable hardware.

6. Improvements over PARC and Domain-Specific Adaptations

Key advances over the original PARC framework include:

• Upwind advection replaces right-difference derivatives, improving stability and eliminating spurious oscillations in low-Mach or weak shock regimes.
• Configurable boundary conditions allow for domain-appropriate padding (e.g., Neumann, Dirichlet), essential for microstructural and interface problems.
• Staged, multi-step curriculum training extends simulation rollouts, with stable predictions achieved an order-of-magnitude longer in weak shock scenarios.

Benchmarking in pore collapse and shear band formation demonstrates:

• 2–5x reduction in temperature RMSE compared to PARC at low velocities.
• Elimination of boundary-related artifacts (e.g., high-temperature "hot pixels").
• Superior performance (RMSE, physical location/width of shear bands) to FNO and Neural ODE models, with particular robustness to extrapolation in moderate shock regimes (Cheng et al., 8 Oct 2025).

7. Extensions and Prospective Directions

PARCv2 establishes a foundation for hybrid numerical–learning approaches in fields requiring robust, generalizable, and efficient simulation of nonlinear PDEs. Suggested future extensions include:

• Incorporation of soft physics constraints (e.g., divergence-free penalties) during or post-training to enforce physical invariants precisely.
• Extension to spatially variable coefficients (heterogeneous media), anisotropic operators, or higher-order equations (e.g., Cahn–Hilliard).
• Generalization of the differentiator–integrator paradigm to stochastic or diffusion-based models outside fluid dynamics (e.g., neural SDEs/ODEs, generative diffusion models).

A plausible implication is that the separation of physics-informed differentiation and data-driven integration can be adapted for other operator-learning frameworks, mitigating spectral bias and improving stability in a wide class of temporally evolved systems (Nguyen et al., 2024, Cheng et al., 8 Oct 2025).