Multiplicative Constraints for Physics-Informed Learning: A Deep Dive into Prime-Weighted Gradient Flow

A comprehensive exploration of the multiplicative PINN framework, from mathematical foundations to real-time Navier–Stokes solutions, with detailed theory, derivations, and pedagogical explanations.

Introduction: The gradient pathology problem

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training (Raissi, Perdikaris, and Karniadakis 2019). However, a fundamental challenge has plagued the field: gradient pathology—the phenomenon where data-fitting and physics-constraint terms create conflicting gradient signals that destabilize training (Wang, Yu, and Perdikaris 2021).

The additive penalty approach and its limitations

Traditional PINNs enforce physics constraints through additive penalty terms:

\[ L_{\text{additive}} = L_{\text{data}} + \lambda_1 L_{\text{physics}_1} + \lambda_2 L_{\text{physics}_2} + \cdots \]

where \(\lambda_i\) are hyperparameters balancing different loss components. While conceptually simple, this approach suffers from several critical issues:

1. Gradient conflicts: The gradient of the total loss becomes \(\nabla L = \nabla L_{\text{data}} + \sum_i \lambda_i \nabla L_{\text{physics}_i}\). When these gradients point in opposing directions, optimization becomes unstable.

2. Hyperparameter sensitivity: The choice of \(\lambda_i\) values dramatically affects convergence, requiring extensive tuning for each problem.

3. Loss landscape distortion: Additive penalties can create sharp valleys and plateaus that trap optimizers, especially when constraints are nearly satisfied.

4. Multi-constraint incompatibility: When multiple constraints are enforced simultaneously, their gradients can cancel or amplify unpredictably, leading to training failures.

Figure 1: Visualization of gradient conflicts in additive loss formulations. When data and physics gradients point in opposing directions, their sum can lead to slow convergence or oscillation.

A multiplicative alternative

This work introduces a fundamentally different approach: multiplicative constraint enforcement. Instead of adding constraint terms, we multiply the data loss by a constraint-dependent factor:

\[ L_{\text{multiplicative}} = L_{\text{data}} \cdot C(v) \]

where \(C(v)\) is a constraint factor that depends on violation scores \(v\). This simple change has profound implications for gradient flow, as we will explore in detail.

Figure 2: Comparison of loss formulations: additive penalties create conflicting gradients while multiplicative scaling preserves gradient direction.

Mathematical foundations: Why multiplication preserves gradient direction

Gradient analysis of additive vs multiplicative losses

To understand why multiplicative constraints avoid gradient conflicts, we must examine how gradients propagate through each formulation.

Additive formulation: For \(L_{\text{add}} = L_{\text{data}} + \lambda L_{\text{constraint}}\):

\[ \nabla_\theta L_{\text{add}} = \nabla_\theta L_{\text{data}} + \lambda \nabla_\theta L_{\text{constraint}} \]

The gradient is a vector sum. If \(\nabla_\theta L_{\text{data}}\) and \(\nabla_\theta L_{\text{constraint}}\) point in opposite directions, they can cancel or create oscillatory dynamics.

Multiplicative formulation: For \(L_{\text{mult}} = L_{\text{data}} \cdot C(v)\):

\[ \nabla_\theta L_{\text{mult}} = C(v) \cdot \nabla_\theta L_{\text{data}} + L_{\text{data}} \cdot \nabla_\theta C(v) \]

Using the product rule, we can rewrite this as:

\[ \nabla_\theta L_{\text{mult}} = C(v) \cdot \nabla_\theta L_{\text{data}} + L_{\text{data}} \cdot \frac{\partial C}{\partial v} \cdot \nabla_\theta v \]

The key insight: Direction preservation

The critical observation is that the first term \(C(v) \cdot \nabla_\theta L_{\text{data}}\) is a scaled version of the original data gradient. The direction remains unchanged; only the magnitude is modulated by \(C(v)\).

When constraints are satisfied (\(v \approx 0\)), we design \(C(v) \approx 1\), so the gradient direction matches the pure data gradient. When constraints are violated (\(v > 0\)), \(C(v)\) grows, amplifying the gradient magnitude while preserving its direction.

The second term \(L_{\text{data}} \cdot \frac{\partial C}{\partial v} \cdot \nabla_\theta v\) provides additional gradient signal to reduce violations, but crucially, it doesn’t conflict with the data gradient direction—it adds a correction term that guides toward constraint satisfaction.

Figure 3: Gradient scaling behavior: multiplicative constraints scale the gradient magnitude while preserving direction, unlike additive methods that alter the gradient direction.

Geometric interpretation

In the loss landscape, additive penalties create new valleys that may be orthogonal or opposed to the data loss valley. Multiplicative constraints, in contrast, modulate the depth of existing valleys without creating new competing minima.

Figure 4: Loss landscape comparison: additive penalties (left) create competing minima, while multiplicative constraints (right) preserve the landscape geometry while enforcing constraints.

The constraint factor: Euler gates and exponential barriers

The constraint factor \(C(v)\) is constructed from two complementary mechanisms: an Euler product gate for attenuation and an exponential barrier for amplification.

Euler product gate: Hierarchical attenuation

The Euler gate uses a truncated product over primes to create a smooth attenuation function:

\[ G(v) = \prod_{p \in \mathcal{P}} \left(1 - p^{-\tau v}\right) \]

where \(\mathcal{P} = \{2, 3, 5, 7, 11, \ldots\}\) is a set of small primes, and \(\tau > 0\) controls the gate sharpness.

Behavior analysis:

• When \(v = 0\) (constraints satisfied): Each term \((1 - p^{-\tau \cdot 0}) = (1 - 1) = 0\), so \(G(0) = 0\). However, in practice, we clamp \(G(v)\) to ensure it never becomes exactly zero (which would eliminate gradients).

• When \(v > 0\) (constraints violated): Each term \((1 - p^{-\tau v}) < 1\), and the product creates a smooth decay. Larger primes contribute smaller corrections, creating a hierarchical structure.

• As \(v \to \infty\): Each term approaches 1, so \(G(v) \to 1\), meaning the gate becomes transparent for severe violations.

Figure 5: Individual prime contributions to the Euler gate. Smaller primes (2, 3) dominate the product, while larger primes provide fine-grained corrections.

Why primes? The logarithmic distribution of primes creates a natural hierarchy. Smaller primes (2, 3, 5) dominate the product, while larger primes (7, 11, 13) provide fine-grained corrections. This mirrors spectral gap hierarchies in graph theory (Chung 1997), where fundamental modes dominate and higher modes refine.

Figure 6: Visualization of prime-number Euler gates showing the hierarchical constraint structure. Each prime creates a unique attenuation pattern, and their product forms a crystalline gate structure that maps constraints to prime factors.

Exponential barrier: Violation amplification

The exponential barrier provides strong amplification when constraints are violated:

\[ B(v) = e^{\gamma v} \]

where \(\gamma > 0\) controls the barrier sharpness.

Behavior analysis:

• When \(v = 0\): \(B(0) = 1\), so no amplification.
• When \(v > 0\): \(B(v) > 1\) grows exponentially, strongly penalizing violations.
• As \(v \to \infty\): \(B(v) \to \infty\), creating an effective hard constraint.

Figure 7: Exponential barrier behavior for different sharpness values γ. Higher γ creates steeper penalties for constraint violations.

Combining gate and barrier

The constraint factor combines both mechanisms:

\[ C(v) = \max(G(v), B(v)) \]

This ensures that: - When constraints are satisfied (\(v \approx 0\)): \(G(v)\) dominates, providing smooth attenuation. - When constraints are violated (\(v > 0\)): \(B(v)\) dominates, providing strong amplification.

The \(\max\) operation preserves the stronger effect at each violation level, creating a dual-mode constraint enforcement mechanism.

Figure 8: Combined constraint factor C(v) = max(G(v), B(v)). The gate dominates near zero, while the barrier dominates for larger violations.

Practical implementation

In practice, we clamp \(C(v)\) to prevent numerical issues:

\[ C(v) = \text{clamp}(\max(G(v), B(v)), \epsilon, M) \]

where \(\epsilon \approx 10^{-6}\) prevents division by zero, and \(M \approx 10^6\) prevents overflow. This ensures stable training across all violation regimes.

The superconducting analogy: Prime-weighted phase coherence

Conceptual framework

The framework draws a conceptual analogy between constraint satisfaction and superconducting phase coherence, inspired by connections between number theory and statistical mechanics (Bost and Connes 1995).

BCS theory and Cooper pairs

In superconductivity, electrons form Cooper pairs that condense into a coherent quantum state with zero electrical resistance (Bardeen, Cooper, and Schrieffer 1957). The critical temperature \(T_c\) marks a phase transition where this coherent state emerges.

The analogy:

Superconductor	Multiplicative Constraint System
Energy Gap \(\Delta(p)\)	Prime Spectral Gap \(\lambda_1(p) \propto 1/\log(p)\)
Cooper Pairing	Euler Product \(\prod(1-p^{-\tau v})\)
Critical Temperature \(T_c\)	Critical \(\beta_c = 1\) (Bost-Connes transition)
Zero Resistance	Zero Violation Rate (0.00%)
Meissner Effect	Gradient Expulsion (No Conflicts)

The Bost-Connes phase transition

At the critical point \(\beta_c = 1\), the Riemann zeta function \(\zeta(\beta)\) diverges:

\[ Z(\beta) = \zeta(\beta) \cdot \text{Tr}(e^{-\beta L}) \]

This arithmetic pole nucleates a phase transition, analogous to how electron-phonon coupling creates Cooper pairs in BCS theory. In our framework, primes provide the “pairing potential” that eliminates gradient scattering.

Figure 9: Prime spectral gap hierarchy: smaller primes create larger gaps, providing stronger constraint enforcement. This mirrors energy gaps in superconducting systems.

Spectral gap hierarchy

The spectral gap of the constraint graph Laplacian controls gradient flow conductance. By weighting constraints with:

\[ w_c = (1 + \log p_c)^{-\alpha} \]

we engineer a hierarchical gap structure: - \(p = 2\) (most important constraint) → largest gap = strongest pairing - \(p = 3, 5, 7, \ldots\) → decreasing gaps = weaker pairings
- Product over all \(p\) → macroscopic quantum coherence across constraints

This creates a “zero-resistance” constraint flow where constraints propagate without dissipation, analogous to superconducting current flow.

Experimental evidence: Zero-resistance state

Our benchmarks demonstrate the “superconducting phase”:

Figure 10: Experimental evidence of the ‘superconducting’ state: dramatic improvements across all constraint types when using multiplicative enforcement.

The analogy, while qualitative, provides valuable intuition for understanding why multiplicative constraints achieve such dramatic improvements over additive methods.

Multi-constraint enforcement: Simultaneous satisfaction

The challenge of multiple constraints

Real-world applications often require enforcing multiple constraints simultaneously: - Monotonicity: Outputs must follow monotonic trends (e.g., pricing functions) - Lipschitz continuity: Gradient magnitudes must be bounded (smoothness) - Positivity: Outputs must remain positive (e.g., probabilities, concentrations) - Convexity: Functions must be convex (optimization guarantees)

Traditional additive methods struggle with multiple constraints because their gradients can conflict, creating training instabilities.

Multiplicative solution: Constraint factorization

The multiplicative framework handles multiple constraints through a factorization approach. Each constraint \(i\) is assigned a prime \(p_i\), and violations are combined multiplicatively:

\[ v_{\text{total}} = \sum_i w_i \cdot v_i \]

where \(w_i = (1 + \log p_i)^{-\alpha}\) are prime-weighted violation scores. The constraint factor then becomes:

\[ C(v_{\text{total}}) = \max\left(\prod_{i} G_i(v_i), B(v_{\text{total}})\right) \]

where \(G_i(v_i) = (1 - p_i^{-\tau v_i})\) are individual gate terms.

Empirical results: Four simultaneous constraints

We tested the framework on a neural network enforcing four constraints simultaneously. The results demonstrate perfect compatibility:

Figure 11: Multi-constraint enforcement results: all four constraints show substantial improvements, with monotonicity achieving perfect satisfaction (100% improvement).

Key observations:

1. Zero gradient conflicts: All four constraints were enforced simultaneously without training instability.

2. Perfect monotonicity: The 100% improvement in monotonicity demonstrates that multiplicative constraints can achieve perfect satisfaction, something impossible with additive methods due to gradient conflicts.

3. Consistent improvements: All constraints showed substantial improvements, indicating the framework scales naturally to multiple constraints.

Figure 12: Monotonicity constraint enforcement: chaotic function with 31.31% violations (top) transforms into perfectly monotonic function with 0.00% violations (bottom). The ‘singularity achieved’ moment demonstrates perfect constraint satisfaction.

Why multiplicative constraints avoid conflicts

The key insight is that multiplicative scaling preserves gradient directions. When multiple constraints are active:

\[ \nabla_\theta L = C(v_{\text{total}}) \cdot \nabla_\theta L_{\text{data}} + L_{\text{data}} \cdot \sum_i \frac{\partial C}{\partial v_i} \cdot \nabla_\theta v_i \]

Each constraint contributes a correction term \(\frac{\partial C}{\partial v_i} \cdot \nabla_\theta v_i\), but these terms don’t conflict with each other or with the data gradient. They form a coherent correction that guides optimization toward the intersection of all constraint manifolds.

Navier-Stokes: The ultimate validation

The Navier-Stokes challenge

The Navier-Stokes equations represent one of the most challenging PDE systems in computational fluid dynamics:

\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f} \]

\[ \nabla \cdot \mathbf{u} = 0 \quad \text{(incompressibility)} \]

where \(\mathbf{u}\) is velocity, \(p\) is pressure, \(\nu\) is kinematic viscosity, and \(\mathbf{f}\) is body force.

Challenges: 1. Non-linearity: The advection term \((\mathbf{u} \cdot \nabla)\mathbf{u}\) creates strong non-linear coupling 2. Incompressibility: The constraint \(\nabla \cdot \mathbf{u} = 0\) must be satisfied exactly 3. Multi-scale dynamics: Turbulent flows exhibit behavior across many spatial and temporal scales 4. Computational cost: Traditional CFD methods require hours or days for high-resolution simulations

Streamfunction architecture: Guaranteed incompressibility

To ensure incompressibility at machine precision, we use a streamfunction architecture:

\[ u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x} \]

where \(\psi\) is the streamfunction. This construction guarantees \(\nabla \cdot \mathbf{u} = 0\) by construction, since:

\[ \nabla \cdot \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial^2 \psi}{\partial x \partial y} - \frac{\partial^2 \psi}{\partial y \partial x} = 0 \]

The neural network outputs \(\psi\) and \(p\), from which velocities are computed via automatic differentiation. This architectural choice eliminates incompressibility violations at the source.

Figure 13: Streamfunction visualization: the neural network outputs ψ, from which divergence-free velocity fields are derived via automatic differentiation.

Multiplicative constraint enforcement

The momentum equations are enforced using multiplicative constraints:

\[ L = L_{\text{data}} \cdot C(v_{\text{momentum}}) \]

where \(v_{\text{momentum}}\) measures violation of the momentum equations. The Euler gate and exponential barrier work together to ensure physics satisfaction while maintaining stable training.

Breakthrough results

Our 2D Navier-Stokes solver achieves unprecedented performance:

• 99.64% residual reduction: PDE residual decreases from 0.0028 to 1×10⁻⁵
• 1,000,908 physics-informed states/second: Real-time inference capability
• 8000 time steps in ~8 ms: Complete simulation in milliseconds
• Machine-precision incompressibility: \(|\nabla \cdot \mathbf{u}| < 10^{-9}\) (guaranteed by architecture)

Figure 14: Performance comparison: multiplicative PINN achieves 745,919× speedup over traditional CFD methods while maintaining physics accuracy.

Real-time Navier-Stokes solution: Traditional CFD (left) requires hours to converge, while the multiplicative PINN (center) generates solutions in milliseconds.

Figure 15: Primary Navier-Stokes visualization showing velocity streamlines and pressure contours. The solution exhibits physically realistic flow patterns with proper vorticity and pressure gradients.

Physics validation: Beyond residuals

Beyond achieving low residuals, we validate that solutions preserve physical structure:

Divergence and vorticity: The divergence field confirms incompressibility (\(|\nabla \cdot \mathbf{u}| < 10^{-9}\)), while vorticity \(\omega = \nabla \times \mathbf{u}\) captures rotational dynamics correctly.

Energy dissipation: Kinetic energy decreases over time following viscous dissipation laws, demonstrating proper physics preservation.

Pressure-velocity correlation: Pressure gradients correlate correctly with velocity changes (90.56% correlation), maintaining the physical relationship between these quantities.

Figure 16: Physics validation: Divergence field (top-left) confirms incompressibility, vorticity field (top-right) shows rotational dynamics, and combined analysis (bottom) validates physical consistency.

Figure 17: Temporal evolution analysis: Energy dissipation (top-left), pressure-velocity correlation (top-right), velocity components (bottom-left), and pressure derivatives (bottom-right) all demonstrate physically consistent behavior over time.

Training dynamics and stability

Lyapunov stability sketch (multiplicative dynamics)

We sketch exponential convergence of constraint violations under the multiplicative gradient flow \(\dot{\theta} = -\nabla L_{\text{mult}}\). Let the constraint manifold be \(\mathcal{M} = \{ \theta : c(\theta) = 0 \}\) and define the violation energy:

\[ V(\theta) = \tfrac{1}{2}\|c(\theta)\|^2, \quad L_{\text{mult}}(\theta) = L_{\text{data}}(\theta) \cdot S(V(\theta)) \]

Along the dynamics,

\[ \dot V = -S(V)\langle \nabla V, \nabla L_{\text{data}}\rangle - L_{\text{data}}(\theta) S'(V)\|\nabla V\|^2 \]

Assume locally: (i) \(L_{\text{data}}(\theta) \ge m > 0\) (or use \(L_{\text{data}}+\epsilon\)), (ii) \(\|\nabla L_{\text{data}}\| \le M\), (iii) a constraint PL inequality \(\|\nabla V\|^2 \ge 2\lambda V\), and (iv) monotone scaling \(S'(V) \ge s_0 > 0\) (true for the exponential barrier and truncated gate).

Then the negative term dominates and yields \(\dot V \le -2k\lambda V\) for some \(k>0\), so:

\[ V(t) \le V(0)\,e^{-2k\lambda t} \]

Hence constraint violation decays exponentially, implying local exponential convergence to \(\mathcal{M}\).

Global convergence (sketch)

To extend beyond local convergence, assume compact sublevel sets of \(L_{\text{mult}}\). This holds if either \(L_{\text{data}}\) is coercive or \(V(\theta)\to\infty\) as \(\|\theta\|\to\infty\) with \(S(V)\) at least linear in \(V\). With \(L_{\text{data}}(\theta)\ge m>0\) and \(S'(V)\ge s_0>0\) for \(V>0\), one can show \(\dot V < 0\) outside a neighborhood of \(\mathcal{M}\), trajectories remain bounded, and LaSalle’s invariance principle yields convergence to \(\mathcal{M}\) from arbitrary initialization.

Rate matching for hyperparameter selection (sketch)

Near \(\mathcal{M}\), use the constraint PL inequality \(\|\nabla V\|^2 \ge 2\lambda_{\min}(J_c J_c^{\top})\,V\). Then:

\[ \dot V \le -2m\,S'(0)\,\lambda_{\min}(J_c J_c^{\top})\,V \]

For \(S(V)=G(V)e^{\gamma V}\), \(S'(0)\approx \gamma G(0) + G'(0)\) (or after clamping, \(S'(0)\approx \gamma\epsilon\)), yielding a rate bound that links \(\gamma\) to the desired exponential decay and provides a predictive tuning rule.

Stochastic mini-batch extension (sketch)

Model mini-batch training as an SDE:

\[ d\theta_t = -\nabla L_{\text{mult}}(\theta_t)\,dt + \sqrt{2\sigma^2}\,dW_t \]

Applying Itô’s formula to \(V\) gives a drift term dominated by \(L_{\text{data}}S'(V)\|\nabla V\|^2\) and a diffusion term \(\sigma^2\mathrm{tr}(\nabla^2 V)\). Under a dissipativity condition, one obtains:

\[ \frac{d}{dt}\mathbb{E}[V] \le -a\,\mathbb{E}[V] + b \]

so \(\mathbb{E}[V(t)]\) contracts exponentially to an \(O(\sigma^2)\) neighborhood. With log-Sobolev or Poincaré conditions, exponential concentration inequalities follow.

Convergence analysis

Training with multiplicative constraints exhibits stable convergence without gradient explosion, even for long unrolled simulations. The key factors contributing to stability are:

1. Gradient direction preservation: Multiplicative scaling doesn’t create conflicting gradient signals
2. Smooth constraint factors: Euler gates provide smooth transitions between constraint regimes
3. Bounded amplification: Exponential barriers are clamped to prevent numerical overflow

Figure 18: Simulated training curves comparing additive vs multiplicative constraint enforcement. Multiplicative methods show stable convergence while additive methods exhibit instability.

Figure 19: Training dynamics from 3D large eddy simulation experiments. The curves show stable convergence without gradient explosion, even for extended training periods.

Robustness across random seeds

We validated robustness by training with five different random seeds (42, 123, 456, 789, 321). Results show consistent performance:

Figure 20: Robustness validation across 5 random seeds. The low variance demonstrates reliable, reproducible training dynamics.

The low variance demonstrates that multiplicative constraints provide reliable, reproducible training dynamics.

Scalability analysis

Performance scales gracefully with problem size:

Figure 21: Scalability analysis: per-point processing time remains nearly constant as grid size increases, demonstrating excellent scalability.

Even at 100×100 resolution, the framework maintains near-constant per-point processing time, demonstrating excellent scalability.

Implementation details

Architecture design

The neural network architecture for Navier-Stokes uses: - Input: Spatial coordinates \((x, y)\) and time \(t\) - Output: Streamfunction \(\psi\) and pressure \(p\) - Architecture: Multi-layer perceptron with 4-5 hidden layers, 64-128 neurons per layer - Activation: Swish/SiLU for smooth gradients - Total parameters: ~33,539 (compact and efficient)

Figure 22: Neural network architecture for Navier-Stokes: inputs (x, y, t) → hidden layers → outputs (ψ, p) → derived velocities (u, v) via autodiff.

Training procedure

1. Data sampling: Collocation points sampled uniformly from domain
2. Loss computation: Multiplicative constraint loss with Euler gates and exponential barriers
3. Optimization: Adam optimizer with learning rate 1e-3 to 1e-4
4. Constraint weighting: Prime-based weights \(w_i = (1 + \log p_i)^{-\alpha}\) with \(\alpha = 1.0\)

Hyperparameter settings

• Gate sharpness: \(\tau = 3.0\) (controls Euler gate decay rate)
• Barrier sharpness: \(\gamma = 5.0\) (controls exponential barrier growth)
• Prime set: \(\mathcal{P} = \{2, 3, 5, 7, 11\}\) (first five primes)
• Clamping bounds: \([\epsilon, M] = [10^{-6}, 10^6]\) (prevents numerical issues)

These settings were found to work well across multiple problem types, though fine-tuning can improve performance for specific applications.

Applications and impact

Engineering applications

Real-time aerodynamics: Aircraft design with instant feedback on aerodynamic properties, enabling rapid iteration cycles.

Interactive CFD: Automotive and aerospace industries can use real-time fluid simulation for design optimization.

Turbine optimization: Wind and hydroelectric turbine design with instant efficiency feedback.

Scientific computing

Climate modeling: High-resolution climate simulations with real-time updates for policy decisions.

Weather prediction: Fast, accurate weather forecasting with physics-informed neural networks.

Biomedical flows: Surgical planning with real-time blood flow simulation.

Autonomous systems

Underwater navigation: Drones with real-time flow awareness for efficient path planning.

Aircraft control: Real-time aerodynamic response for flight control systems.

Manufacturing: Process flow optimization with guaranteed physics constraints.

Limitations and future directions

Current limitations

1. Hardware validation: Benchmarks reported on specific hardware; broader cross-platform validation needed.

2. Theoretical foundations: The superconducting analogy is conceptual; formal theoretical analysis of convergence and optimality would strengthen the framework.

3. Baseline comparisons: While we compare against additive PINNs, comparisons with other constraint enforcement methods (e.g., augmented Lagrangian, interior point methods) would provide broader context.

4. PDE family coverage: While validated on Navier-Stokes, Poisson, and heat equations, broader validation across PDE families (e.g., wave equations, Schrödinger equations) is ongoing.

Future research directions

Theoretical advances: - Formal convergence analysis for multiplicative constraints - Optimal prime selection strategies - Connection to optimal transport and Wasserstein distances

Methodological improvements: - Adaptive constraint weighting based on violation severity - Hierarchical constraint decomposition for complex systems - Integration with uncertainty quantification

Applications: - Multi-physics coupling (fluid-structure interaction, magnetohydrodynamics) - High-dimensional PDEs (4D+ spatiotemporal systems) - Real-time deployment on embedded systems (FPGAs, edge devices)

Conclusion

Multiplicative constraints represent a paradigm shift in physics-informed machine learning. By replacing additive penalties with multiplicative scaling, we preserve gradient directions while enabling stable, efficient constraint enforcement.

Key contributions

1. Theoretical foundation: Mathematical analysis showing why multiplicative constraints preserve gradient directions and avoid conflicts.

2. Prime-weighted hierarchy: Euler product gates create structured constraint enforcement with natural hierarchy.

3. Multi-constraint compatibility: First framework to successfully enforce multiple simultaneous constraints without gradient conflicts.

4. Real-time Navier-Stokes: Breakthrough performance enabling real-time fluid simulation with physics accuracy.

5. Superconducting analogy: Conceptual framework connecting constraint satisfaction to phase coherence phenomena.

Broader implications

The multiplicative constraint framework opens new possibilities for: - Reliable AI systems: Safety-critical applications requiring guaranteed constraint satisfaction - Real-time simulation: Interactive design and control systems with instant physics feedback
- Scientific discovery: Fast exploration of parameter spaces with physics guarantees - Democratized simulation: Making high-fidelity physics simulation accessible to broader communities

As physics-informed machine learning continues to mature, multiplicative constraints provide a robust foundation for building reliable, efficient, and scalable systems that respect physical laws while achieving unprecedented performance.

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Acknowledgments: This work builds on foundational research in physics-informed neural networks (Raissi, Perdikaris, and Karniadakis 2019), gradient pathology analysis (Wang, Yu, and Perdikaris 2021), spectral graph theory (Chung 1997), and connections between number theory and statistical mechanics (Bost and Connes 1995; Bardeen, Cooper, and Schrieffer 1957).

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Code & Citation

Source Code: The complete implementation is available on GitHub at https://github.com/sethuiyer/multiplicative-pinn-framework

If this work helped your research, please consider citing:

Sethurathienam Iyer (2025). “ShunyaBar: Spectral–Arithmetic Phase Transitions for Combinatorial Optimization”. Zenodo. doi:[10.5281/zenodo.18214172](https://doi.org/10.5281/zenodo.18214172)

@software{iyer2025shunyabar,
  author       = {Sethurathienam Iyer},
  title        = {{ShunyaBar: Spectral–Arithmetic Phase Transitions 
                   for Combinatorial Optimization}},
  year         = 2025,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18214172},
  url          = {https://doi.org/10.5281/zenodo.18214172}
}