Examples

Experiment Evidence and New Equilibria

• Bayesian, Joint, Dynamic, and Certified Examples
• Evidence and Protocol Example

The examples are compact CPU-first validations of the public package API. They are not benchmark scripts; they are compact checks that show how to assemble a SILVA equilibrium, train it, inspect residuals, and move the same pattern to a larger experiment.

Run All Examples

python examples/scalar_deq.py
python examples/graph_silva.py
python examples/vision_channels.py
python examples/molecules.py
python examples/custom_layers.py
python examples/deq_engine_bridge.py
python examples/add_layers_on_top.py
python examples/cortex_hierarchy.py
python examples/spatial_cortex.py
python examples/point_architecture_catalog.py
python examples/scientific_operators.py
python examples/frontier_equilibria.py
python examples/advanced_equilibria.py
python examples/optical_flow_silva.py
python examples/constrained_optimization.py
python examples/stacked_architecture.py
python examples/datasets_quickstart.py
python examples/paper_family_cases.py
python examples/raft_deq_flow.py
python examples/source_data_families.py

What Each Example Covers

Example	Package objects	Main check
Scalar DEQ	fixed_point, full_jacobian, stability_report	numerical solution equals a closed-form fixed point
Graph SILVA	SILVAGraphLayer, SolverConfig	graph state shape, gradients, stability report
Vision Channels	SILVAImageLayer	image tensor state and residual solve
Molecules	SILVAGraphLayer	atoms as entities, bonds as edges, molecule pooling
Custom Layers	SILVALayer, custom nn.Module branches	replacing interaction branches
DEQ Engine Bridge	SILVADEQEngine, SILVADEQConfig, silva_deq	solving arbitrary single-state and multi-state systems
Cortex Hierarchy	SILVACortexLayer, SILVACortexNetwork	deep internal modules inside linked equilibrium points
Spatial SILVA Cortex	SILVACortexLayer, silva_equilibrium_model	residual CNN and U-Net inside one point linked to a different vector point
Point Architecture Catalog	point architecture registry and SILVACortexLayer	ten shape-preserving vector, token, spatial, gradient, and tiny-data checks
Full Cortex Operators	every SILVACortexLayer slot and all 25 branch factory names	internal sequence, self, local, global, custom, output, normalization, solver, shapes, and gradients
Scientific Operators	ODE flow, implicit PDE steps, reaction-diffusion, Burgers, Fourier operators, graph PDEs	analytic errors, fixed-point residuals, boundaries, gradients, and resolution changes
Recent Equilibrium Families	Fourier equilibrium, physics graph, homotopy, and empirical-measure SILVA models	four bounded reproductions with solver or discrepancy diagnostics
Advanced Equilibria	monotone graph, injected transformer, Poisson mirror, physics-informed ODE, DAE root, residual objective	six equation-checked mechanisms with distinct numerical and task diagnostics
Optical Flow SILVA	SILVADEQFlow, RAFT-style correlation helpers	synthetic flow fixed point, EPE, smoothness, gradients
Constrained Optimization	SILVAProjectedQPLayer, silva_projected_qp_layer	projected-QP fixed point, simplex constraints, gradients
Stacked Architecture	SILVAGraphNetwork, mixed solvers	multi-layer equilibrium stack on a selected device
Dataset Quickstart	dataset loaders and adapters	public data to GraphTensorBatch to model
Citation-Aware Reporting	presets, solvers, diagnostics, citation audit	methods sentence and citation checklist for a concrete configuration
Paper Family Cases	sequence DEQ, MDEQ, IGNN, INR, DEQ-DDIM	compact architecture and gradient smokes across generalized cases
RAFT and DEQ-Flow	SILVARAFTDEQ, correction loss, cached state	coupled flow solve, sparse correction predictions, backward pass, and reuse
Source-Data Families	source receipts, verified snapshots, monDEQ, pcDEQ, NEMON, EIGNN, MGNNI, DeltaDEQ	six real-tensor mechanism and gradient checks with explicit non-benchmark scope

Shared Pattern

Every example follows the same engineering path:

\[ \text{raw object} \to (x,E,b,y) \to z^\star=f_\theta(z^\star,x) \to \hat y=R_\phi(z^\star) \to \text{diagnostics}. \]

The same pattern supports user datasets after preprocessing into the package tensor contract.

Executable Evidence Map

Every worked page now retains its introductory route and adds the complete program, measured compact output, mathematical result contract, interpretation, and full-scale transfer record.

Worked page	Executable program	Compact evidence
Advanced Expansions	examples/advanced_expansions.py	Posterior variance, coupled-root residual, trajectory shape, and certified margins
Advanced Equilibria	examples/advanced_equilibria.py	One result for each advanced equilibrium family, with family-specific residuals
Citation Aware Reporting	examples/reproduction_registry.py	A complete machine-readable source and verification record
Constrained Optimization	examples/constrained_optimization.py	Simplex feasibility, energy, solver residual, and parameter gradients
Cortex Hierarchy	examples/cortex_hierarchy.py	Per-point state shapes, solver choices, logits, loss, and gradients
Custom Layers	examples/custom_layers.py	The custom state shape, final residual, and differentiable loss path
Datasets Quickstart	examples/datasets_quickstart.py	Dataset identity, tensor shape, and measured classification accuracy
Deq Engine Bridge	examples/deq_engine_bridge.py	State shape, iterations, residual ratio, and gradient availability
Emerging Equilibria	examples/emerging_equilibria.py	Family-specific exact-solution, boundary, reconstruction, or trajectory checks
Evidence And Protocols	examples/evidence_and_protocols.py	Repeated-seed statistics, residuals, fingerprints, and three explicit scale tiers
Frontier Equilibria	examples/frontier_equilibria.py	Task, equation, invariance, and fixed-point residuals for four operator classes
Full Cortex Operators	examples/full_cortex_operators.py	Branch activations, solver history, state shape, loss, and gradients
Full Scale Training	examples/add_layers_on_top.py	A measured training loss from the complete optimization path
Graph Silva	examples/graph_silva.py	Node-state shape, task loss, equilibrium residual, and gradients
Learned Solvers	examples/learned_solvers.py	Teacher and learned-solver residuals plus exact, jfb, and shine gradients
Molecules	examples/molecules.py	Atom and molecule tensor shapes, residual, prediction loss, and gradients
Optical Flow Silva	examples/optical_flow_silva.py	Flow shape, endpoint error, iterations, residual, and gradients
Paper Family Cases	examples/paper_family_cases.py	Shape and residual checks across sequence, vision, graph, and diffusion cases
Point Architecture Catalog	examples/point_architecture_catalog.py	Parameters, loss, residual trajectory, and gradient norm for every architecture
Quantum Deq	examples/quantum_deq.py	Circuit measurements, equilibrium residual, task output, and circuit gradients
Raft Deq Flow	examples/raft_deq_flow.py	Flow shape, correction trajectory, solver residual, loss, and gradients
Reproduction Registry	examples/reproduction_registry.py	Verification levels, preserved mechanisms, scale tiers, and source obligations
Scalar Deq	examples/scalar_deq.py	Closed-form agreement, final residual, iteration count, and implicit gradient
Scientific Operators	examples/scientific_operators.py	Ode error plus pde, boundary, and equilibrium residuals
Source Data	examples/source_data_families.py	Losses, certificates, residuals, scale allocation, and cache activity on source data
Spatial Cortex	examples/spatial_cortex.py	Spatial and vector state shapes, per-point solvers, loss, and gradients
Stacked Architecture	examples/stacked_architecture.py	Logit shape, pointwise solvers, loss, and gradient flow across the stack
Structured Equilibria	examples/structured_equilibria.py	Certificates, positivity, one-sided bounds, scale weights, and cache activity
Vision Channels	examples/vision_channels.py	Image-state shape, iteration count, residual, loss, and gradients

Across the collection, a reported result is treated as the tuple

\[ \mathcal E = (\text{task metric},\text{fixed-point residual},\text{iterations}, \text{gradient evidence},\text{domain invariants}). \]

Keeping these entries separate prevents task quality, solver convergence, and structural validity from being collapsed into one number.

Where to Go Next

Question	Page
How do I choose an example for my data and state layout?	Case Atlas
Where should a first-time reader begin?	Introduction by Example
How can I run the examples and validation suite?	Run Everything