Start Here

This path goes from first principles to a usable PyTorch model.

First Hour

1. Open Run Everything and verify the install.
2. Read Derivation Workbook through the scalar and graph-local sections.
3. Run Equation-to-Code Walkthrough.
4. Open Selecting Model Families to choose a SILVA, DEQ, MDEQ, flow, or optimization family from one factory.
5. Open Method Adaptation Atlas to map external methods to SILVA APIs and citations.
6. Open Paper Family Adaptations for the DEQ, MDEQ, Jacobian, TorchDEQ, IGNN, INR, diffusion, RAFT, and DEQ-Flow map.
7. Read Neural Operators, ODEs, PDEs, and SILVA when the state is a dynamical or spatial field.
8. Open Introduction by Example.
9. Read Data Objects and Batching.
10. Run python examples/scalar_deq.py and python examples/graph_silva.py.
11. Open Full-Scale SILVA when the compact path is working and you are ready to choose sharding, precision, accumulation, matrix-free operators, or distributed execution.
12. Use Real-Dataset Reproduction to move from known-solution checks to attributed source subsets and complete local dataset protocols.
13. Open the Family Reproduction Dossiers to inspect the governing equation, replaceable modules, compact evidence, source data, acceptance checks, and editable scale plan for each of the 50 families.
14. Use Learned Solvers and Backward Approximations to separate learned forward acceleration from exact implicit, JFB, and shared-inverse gradient paths.
15. Use Quantum Equilibria to construct a measured circuit transition, replace its backend, and move from compact statevectors to a source-scale experiment.

The main equation is

\[ z^\star=f_\theta(z^\star,x). \]

The main computational check is

\[ \|f_\theta(z^\star,x)-z^\star\|_2. \]

Core Learning Path

Stage	Page	Skill
Fixed-point intuition	Fixed Points	understand residuals and damping
Guided derivation	Derivation Workbook	derive scalar, vector, graph, global, data, and diagnostic equations
Run path	Run Everything	execute examples, notebooks, tests, docs, and data adapters
Mathematical review	Mathematical Foundations	vectors, matrices, norms, derivatives, graph notation
SILVA construction	SILVA From Scratch	build \(S,L,G,H\) terms and solve the layer
Family selection	Selecting Model Families	choose SILVA, DEQ, MDEQ, flow, and optimization modules from one selector
Method adaptation	Method Adaptation Atlas	translate external implicit-layer, DEQ, ODE, optimization, and flow sources into SILVA equations
SILVA operators	SILVA Operators	vary Figure 1 branches and ablations
Cortex hierarchy	Cortex Hierarchies	build linked SILVA points with independently configured MLP, convolutional, U-Net, attention, or graph internals
Internal architecture selection	Point Architecture Catalog	choose among ten vector, token, and spatial fields and compose them inside or across points
Scientific models	Neural Operators, ODEs, PDEs, and SILVA	derive explicit flow, implicit time stepping, PDE residuals, FNO fields, and graph discretizations
Learned forward and backward paths	Learned Solvers and Backward Approximations	train HyperDEQ controls and compare exact implicit, JFB, and SHINE gradients
Quantum equilibrium models	Quantum Equilibria	derive encodings, gates, measurements, fixed points, gradients, and circuit replacement
Cross-family mechanism map	Equilibrium Expansion Atlas	distinguish transition, solver, gradient, objective, and evaluation axes
Scale and reproduce	Full-Scale SILVA	move all 64 families from equation checks to benchmark-ready execution
Source-data reproduction	Real-Dataset Reproduction	verify source receipts and move from compact real subsets to official full splits
Family experiment design	Family Reproduction Dossiers	follow six explicit stages from tensor contracts to a complete cited protocol or declared extension
New family construction	Advanced Extension Handbook	validate primitive modules, public composition equivalence, gradients, serialization, data, and scale registration
Failure analysis	Failure Diagnostics and Recovery	distinguish slow, oscillatory, expansive, stalled, constrained, and backward-solve failures
Stability	Jacobians and Stability	compute Jacobians, products, spectral-radius diagnostics
Dataset adaptation	Datasets and Preprocessing	convert public or private data into the engine

Package Path

Task	Page
Install package and extras	Installation
Pick a solver	Solvers API
Train a learned solver	Learned Solver API
Build a quantum equilibrium	Quantum Equilibria API
Pick a layer	Layers API
Build cortex hierarchies	Architectures API
Choose an internal point architecture	Point Architectures API
Build ODE, PDE, or learned operator models	Scientific Operators API
Pick a model family	Selecting Model Families
Use reference presets	SILVA Presets API
Use the DEQ engine	DEQ Engine API
Use optical-flow utilities	Optical Flow API
Use constrained optimization layers	Optimization API
Adapt datasets	Datasets API
Load attributed source data	Source Data API
Check GPU behavior	Devices API
Configure full-scale execution	Scaling API
Inspect family experiment contracts	Research Depth API
Run common-task family comparisons	Compact Benchmarks API

Long-Form Study

The book and solutions manual Planned will carry the long derivations and solved exercises. The notebooks, Derivation Workbook, and Solver Derivation Lab connect the derivations to executable code today.

The loop used throughout the suite is:

\[ \text{derive} \quad\to\quad \text{implement} \quad\to\quad \text{solve} \quad\to\quad \text{diagnose} \quad\to\quad \text{extend}. \]

The attributed source-data route is developed in Real-Dataset Reproduction.

Where to Go Next

Question	Page
How do I install the package and optional components?	Installation
Can I begin with one small executable example?	Introduction by Example
Which SILVA case matches my problem?	Case Atlas
How can I validate the complete repository?	Run Everything