Differentiable Physics at Machine Precision
KenCarp4 IMEX solvers + SIREN neural layers resolve stiff reaction-diffusion dynamics. Every Jacobian is verified against complex-step differentiation to 5.68 × 10⁻¹⁴.
import axiom.math as am
from axiom.solvers import DifferentiableSolver, KenCarp4
# Axiom Engine: Stiff ADR solver @ 10⁻¹⁴
def reaction_diffusion(t, y, args):
D, k = args
laplacian = am.laplacian_1d(y)
return D * laplacian + k * y * (1 - y)
sol = DifferentiableSolver(
method=KenCarp4(l_stable=True),
adjoint_memory='O(1)',
precision='float64'
).solve(
terms=reaction_diffusion,
t_span=(0, 1),
y0=y_init,
args=(D, k)
)IMEX Stiff Solvers
KenCarp4 implicit-explicit time-stepping engine for L-stable reaction-diffusion coupling.
SIREN Neural Layers
Sinusoidal activations (ω₀ = 30.0) overcome spectral bias, capturing sharp reaction fronts.
Glass-Box Output
Every prediction is traceable to physical law. Regulatory-grade explainability for FDA/FinTech.
Bitwise Determinism
Max Diff = 0.0 across CPU runs. No stochastic seeds, no floating-point non-determinism.
Continuous Graph Neural Diffusion: O(1) Memory
Replaces discrete GNN layers with continuous reaction-diffusion physics. The Adjoint Sensitivity Method divorces memory from integration depth — scaling to billion-node networks.
O(L) Memory Wall → Shattered
Standard deep GNNs cache all intermediate activations. Laminar-GND solves an augmented ODE backwards in time, achieving O(1) memory independent of solver depth.
Reaction-Diffusion Equilibrium
Balances Laplacian diffusion (smoothing) with a learned neural reaction term (energy injection), preventing catastrophic over-smoothing.
AOT-Compiled Sparse Ops
Graph Laplacian operates in BCOO sparse format — never materialised as dense. Guarantees O(E) spatial memory scaling.
Architectural Comparison
| Architecture | Temporal | Spatial Operator | Memory | Over-smoothing |
|---|---|---|---|---|
| Standard GCN | Discrete | Adjacency Matrix | O(L·N·F) | High |
| Transformer (ViT) | Discrete | Dense Self-Attention | O(L·N²) | Moderate |
| Laminar-GND | Continuous | Sparse Laplacian | O(1) w.r.t Depth | Zero |
Homomorphic Encryption: Compute on Encrypted Data
TenSEAL CKKS integration enables encrypted dot-products between proprietary reaction constants and plant state. IP never touches RAM in plaintext.
CKKS Encrypted Inference
Encrypted vs. plaintext divergence: ~9.5 × 10⁻⁷. Acceptable for inference, never used for solver-critical computations.
OPC-UA Industrial Bridge
Asynchronous data ingestion at 100Hz (10ms latency). Connects directly to factory PLCs and DCS systems via the OPC-UA standard.
Constrained MAS: Zero-Hallucination Orchestration
We wrap our deterministic physics solvers in production-grade Multi-Agent System orchestration. LLM agents handle complex data extraction and logistics routing, but are bound by strict data contracts and human-in-the-loop gates.
Agent-Based Routing
Specialised LLM agents handle complex logistics routing, risk assessment, and financial underwriting — each constrained to its domain of competence.
Pydantic Data Contracts
Every agent output is validated against strict Pydantic schemas before propagation. Malformed or hallucinated data is rejected at the boundary — never reaching downstream solvers.
Human-in-the-Loop Gates
Critical decisions require explicit human approval via HITL checkpoints. The AI handles the workflow; the Axiom Engine handles the math — guaranteeing deterministic outcomes.
Why this matters: Standard LLM deployments hallucinate 3–15% of outputs. In financial underwriting and physical infrastructure, that is catastrophic. Our MAS architecture ensures every AI-generated recommendation is either mathematically verified by Axiom or rejected — achieving zero hallucinations in critical decision paths.
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