Equation and PDF Audit
This page records what was checked inside the package documentation surface. It
does not modify the article, book source, or production code outside this
package.
Audit Date
| Field |
Value |
| Package folder |
silva-networks |
| Reviewed on |
August 3, 2026 |
| SILVA article |
arXiv:2607.28989 |
| Companion book and solutions manual |
Planned long-form learning assets |
| Public PDF links |
Local SILVA article PDF, arXiv article links, and package documentation links |
| Article PDF source |
docs/assets/papers/silva-networks-arxiv-2607.28989.pdf |
| PDF metadata validation |
pdfinfo reports 45 pages, title, author, and no encryption |
| Local PDF workspace |
Book/manual drafts remain excluded from public documentation links |
The public documentation keeps the book/manual chapter map visible as a roadmap,
includes the SILVA article PDF for offline reading, and points readers to the
package-native learning path.
Equation Inventory
The package documentation now treats every implemented family as an equation
family with a code target, tutorial target, notebook target, and test target.
| Equation family |
Canonical equation |
Package objects |
Where to study it |
| Fixed point |
\(z^\star=f_\theta(z^\star,x)\) |
fixed_point, DEQLayer, SILVADEQEngine |
Fixed Points, Solver Derivation Lab |
| Damped solve |
\(z_{k+1}=(1-\alpha)z_k+\alpha f_\theta(z_k,x)\) |
picard, SolverConfig.alpha |
Mathematical Foundations |
| SILVA field |
\(f_\theta(z,x)=\Phi(S_\theta(x)+L_\theta(\chi(z),E)+G_\theta(\chi(z),b)+H_\theta(\chi(z)))\) |
SILVALayer, SILVAGraphLayer, presets |
Implementation Derivations |
| Anderson acceleration |
\(\min_c\|Rc\|_2^2+\lambda\|c\|_2^2,\;1^\top c=1\) |
anderson |
Solver Derivation Lab |
| Broyden update |
\(B_{k+1}=B_k+\frac{(s_k-B_ky_k)s_k^\top B_k}{s_k^\top B_ky_k}\) |
broyden |
Solver Derivation Lab |
| GMRES adjoint |
\((I-J_T(z^\star)^\top)u=g\) |
gmres, implicit_adjoint_solve |
Implicit Backward Guide |
| Jacobian stability |
\(\rho((1-\alpha)I+\alpha J_f(z^\star))<1\) |
spectral_radius, stability_report |
Jacobians and Stability |
| Hutchinson penalty |
\(\mathbb E_v\|J_f(z^\star)v\|_2^2\) |
hutchinson_jacobian_norm, silva_jacobian_regularization_loss |
Implicit Layers Bridge |
| Graph local field |
\(L_i=\sum_{j\in N(i)}a_{ij}W_v z_j\) |
GraphLocal, GraphAttentionLocal, TopKLocal |
Case Atlas |
| Global set field |
\(G_i=\gamma_i\odot \psi(\operatorname{pool}_{b_i}\phi(z))\) |
MeanFieldGlobal, GatedMeanFieldGlobal, attention branches |
SILVA Operators |
| Projected QP |
\(z^\star=\Pi_C[z-\eta(Az-B_\theta x-c)]\) |
SILVAProjectedQPLayer |
Optimization API |
| Optical flow |
\(u^\star=T_\theta(u^\star,I_1,I_2,C)\) |
SILVADEQFlow, correlation and warp helpers |
Optical Flow API |
Companion-To-Package Map
| Planned companion material |
Package adaptation available now |
| Fixed-point thinking and contractions |
solvers.py, layers.py, Fixed Points |
| Picard, damping, Anderson, Broyden |
picard, anderson, broyden, Solver Derivation Lab |
| Implicit differentiation and adjoints |
implicit_adjoint_solve, Implicit Backward Guide |
| DEQ and MDEQ cases |
implicit.py, deq_engine.py, bridge notebooks |
| SILVA structured interaction field |
layers.py, presets.py, case atlas, implementation derivations |
| Jacobian regularization and diagnostics |
jacobian.py, diagnostics.py, bridge notebook 06 |
| Vision and molecular examples |
presets.py, flow.py, public experiments, example pages |
Findings
| Finding |
Status |
| Article arXiv metadata is now known. |
Updated docs, notebooks, generator scripts, CFF, README, and BibTeX to arXiv:2607.28989. |
| BibTeX was scattered across pages. |
Centralized in docs/assets/bib/silva-networks.bib and linked from reference pages. |
| Future notebook regeneration could reintroduce stale citation text. |
Generator scripts were updated to the public arXiv citation. |
| Package docs needed an explicit equation audit. |
This page maps equation families to implementation and learning targets. |
| Article access belongs with the canonical arXiv record and the public package docs. |
The docs link the local article PDF, arXiv abstract page, arXiv PDF, DOI, and BibTeX. |
| Book and solutions manual are planned learning assets. |
Public pages point readers to notebooks, derivation pages, and examples. |
Reader Reproduction Path
- Read Paper and References for citation details.
- Open Mathematical Foundations for the
fixed-point, damping, adjoint, and stability derivations.
- Use Solver Derivation Lab to derive the
concrete solver updates used by the package.
- Use Implementation Derivations to map
the symbolic SILVA field into package classes.
- Run Run Everything to execute tests, notebook validation
checks, release checks, and docs builds.
Where to Go Next