physics-guided-neural-network

name	physics-guided-neural-network
description	Design neural networks that embed physical constraints (equations, symmetries, conservation laws) directly into the computational graph. Use when modeling physical systems, scientific computing, or when physics-informed AI is needed. Keywords: PGNN, physics-guided NN, physics-informed ML, physics-embedded NN, scientific ML, holographic QCD, AdS Dirac equation.

Physics-Guided Neural Networks (PGNN)

Design patterns for embedding physical constraints into neural network architectures, ensuring outputs satisfy physical laws while maintaining learning flexibility.

Core Concepts

1. Physics Embedding Strategy

Key Principle: Physical equations are not just loss functions - they are embedded into the network architecture as computational graph constraints.

Three Levels of Integration:

Level	Method	Example
Soft Constraint	Physics loss term	L = L_data + λ L_physics
Hard Constraint	Architecture design	Conservation law enforced
Embedded Equation	Computational graph	Differential equation in forward pass

Design Rule:

Physical Law -> Computational Graph Node
Network Parameters -> Physical Parameters
Output -> Physics-Satisfying Solution

2. Holographic QCD Pattern (from 2604.02906)

AdS/CFT Correspondence in Neural Networks:

• 5D AdS₅ Dirac equation embedded in network
• String diffusion kernel as transformation layer
• Proton mass constraint: M_p ≡ 938 MeV
• Physical parameters guide network weights

Architecture Template:

Input: Experimental data (F_2 structure function)

Physics-Guided Layers:
  1. AdS₅ geometry encoding (spatial embedding)
  2. Dirac equation solver (differential layer)
  3. Diffusion kernel transform (propagation layer)
  
Physical Constraints:
  - Proton mass bound
  - Energy conservation
  - Gauge symmetry

Output: Physics-constrained predictions

3. Symmetry Preservation Patterns

Symmetry Types to Embed:

Symmetry	Embedding Method	Example
Gauge	Equivariant layers	U(1), SU(N)
Translational	Convolution	Periodic systems
Rotational	Spherical harmonics	Molecular systems
Lorentz	Tensor structure	Field theory

Implementation:

# Example: Embedding gauge symmetry
class GaugeEquivariantLayer(nn.Module):
    def forward(self, x):
        # Ensure output transforms same as input under gauge group
        return gauge_transform(self.linear(x))

Implementation Checklist

Step 1: Identify Physical Laws

1. List governing equations (ODE/PDE)
2. Identify conserved quantities
3. Determine symmetries and invariances
4. Note physical parameter bounds

Step 2: Choose Embedding Level

1. Soft: For approximate physics (training guidance)
2. Hard: For exact physics (output guarantee)
3. Embedded: For physics-first (architecture priority)

Step 3: Design Architecture

1. Map physics to network structure
2. Create physics-guided layers
3. Implement constraint modules
4. Define physics loss terms

Step 4: Validate Physics

1. Check conservation laws at output
2. Verify symmetry transformations
3. Test parameter bounds
4. Compare to analytical solutions

Example Architectures

Example 1: Holographic QCD Proton Model

Physics: Quantum Chromodynamics in non-perturbative regime
Equations: AdS₅ Dirac equation, string diffusion kernel
Constraints: Proton mass M_p = 938 MeV

Architecture:
  Input: Bjorken-x, Q² (kinematic variables)
  
  Geometry Encoding:
    - AdS₅ spatial coordinates embedding
    - Warp factor integration
  
  Physics-Guided Core:
    - Dirac equation solver (differential layer)
    - Diffusion kernel transform
    - Holographic mapping
  
  Output: Structure function F₂(x, Q²)
  
  Physical Validation:
    - Mass constraint check
    - Regge limit behavior
    - Deep inelastic scaling

Benefits:
  - Predictions satisfy QCD physics
  - Works in transition regime (no pure theory)
  - Extrapolates to unmeasured regions

Example 2: Fluid Dynamics Simulator

Physics: Navier-Stokes equations
Equations: Conservation of mass, momentum, energy
Constraints: Incompressibility (div v = 0)

Architecture:
  Input: Initial velocity field
  
  Conservation Layers:
    - Mass: divergence-free enforcement
    - Momentum: pressure-velocity coupling
    - Energy: dissipation modeling
  
  Physics-Guided Propagation:
    - Advection term (convective layer)
    - Diffusion term (viscous layer)
    - Pressure correction
  
  Output: Time-evolved velocity field
  
  Physical Validation:
    - Zero divergence check
    - Energy conservation
    - Boundary condition satisfaction

Benefits:
  - Exact conservation laws
  - Stable long-time evolution
  - No spurious numerical artifacts

Example 3: Quantum Field Theory

Physics: Topological field theory (BKT transition)
Equations: Neural network field theory
Constraints: Topological quantum numbers

Architecture:
  Input: Temperature, field configuration
  
  Topological Encoding:
    - Discrete quantum numbers as parameters
    - Vortex/anti-vortex detection
    - Spin-wave modes
  
  Field Theory Layers:
    - Statistical ensemble of fields
    - Network architecture as field definition
    - Parameter density as field measure
  
  Output: Correlation functions, critical points
  
  Physical Validation:
    - BKT transition recovery
    - T-duality verification
    - Spin-wave critical line

Benefits:
  - Recovers known physics exactly
  - Extends to unexplored regimes
  - Topological structure preserved

Common Patterns from Literature

Pattern: Differential Equation as Layer

From: Physics-Informed Neural Networks (PINNs) Key Insight: ODE/PDE can be forward pass operations Application: Use autograd for automatic physics satisfaction

Pattern: Conservation as Architecture

From: Symplectic neural networks Key Insight: Hard constraints > soft constraints for physics Application: Design layers that conserve by construction

Pattern: Symmetry as Transformation

From: Gauge-equivariant neural networks Key Insight: Network should respect group structure Application: Use equivariant layers for symmetry groups

Key Papers Reference

• Physics-Guided NN for Holographic QCD (2604.02906): PGNN with AdS₅ Dirac equation
• Topological Effects in Neural Network Field Theory (2604.02313): Topological quantum numbers
• PINNs (Raissi et al. 2019): Physics-informed neural networks
• Equivariant NN (Cohen & Welling 2016): Group theory in architectures

Tools Used

• exec: Run physics simulations, solve ODEs
• read: Load physical equations, domain knowledge
• write: Document physics-guided architectures
• edit: Modify network configurations

Error Handling

Physics Violation

Symptom: Output violates conservation law Solution: Strengthen hard constraint embedding

Numerical Instability

Symptom: Physics-guided layer diverges Solution: Add regularization, check discretization

Over-constrained Network

Symptom: Network cannot learn due to too many constraints Solution: Relax some constraints to soft penalties

Related Skills

• quantum-classical-hybrid-nn: Quantum computing integration
• neural-dynamics-universal-translator: Neural dynamics modeling
• pinn-neuronal-parameter-estimation: PINN for neuron models

Notes

• Physics embedding is architecture-first, not loss-first
• Hard constraints guarantee physics but may limit flexibility
• Soft constraints allow learning but may violate physics
• Validate against analytical solutions when available
• Consider computational cost of physics layers