silva_psi_gnn Reproduction Dossier

mixed-boundary Poisson graph equilibrium. This dossier connects the source mechanism to its SILVA implementation, compact evidence, replaceable components, and source-scale route. Existing tests and notebooks remain the executable authority.

Evidence boundary

The mechanism is compact-verified in the package suite. The final source-scale stage remains planned until the cited data, complete optimization budget, checkpoints, and evaluation protocol have actually run.

Identity and Sources

Field	Value
Domain	graphs and distributed systems
Task contract	unstructured coordinates, typed nodes, directed edges, forcing, and boundary data -> nodal solution
Source relation	paper-adaptation
References	[60]
Repositories	https://arxiv.org/abs/2302.10891
Editable scale plan	experiments/reproduction/configs/silva_psi_gnn.json

Governing Equation

The domain-level state contract is

\[ Z^\star=T_\theta(Z^\star;X,A,E,b),\qquad \widehat Y=Q_\psi(Z^\star). \]

The implementation registry specializes it operationally as

H_star=h_theta(H_star,G); U_hat=D(H_star); L_res=MSE(A U_hat-B)

Define the root residual

\[ R_\theta(z;x)=z-T_\theta(z;x). \]

At a regular equilibrium, differentiating \(R_\theta(z^\star;x)=0\) gives

\[ \frac{\partial z^\star}{\partial x} = \left(I-\frac{\partial T_\theta}{\partial z}\right)^{-1} \frac{\partial T_\theta}{\partial x}. \]

This identity explains why the forward residual, the conditioning derivative, and the adjoint linear solve must be diagnosed separately from the task metric.

What Is Preserved

• encode-process-decode graph equilibrium
• separate interior incoming/outgoing and Neumann incoming messages
• fixed Dirichlet latent values, Broyden root solving, PDE residual, and Jacobian stabilization

What Can Be Replaced

Each item below is an explicit control rather than an undocumented modification:

• replace encoder, typed message maps, update maps, decoder, or root solver
• supply finite-element matrices only to the physics residual loss
• mesh nodes
• edge count
• latent width
• root-solver budget

Constructor and Shape Contract

silva_psi_gnn(state_dim: 'int', *, coordinate_dim: 'int' = 2, encoder: 'nn.Module | None' = None, forcing_encoder: 'nn.Module | None' = None, processor: 'nn.Module | None' = None, decoder: 'nn.Module | None' = None, config: 'SolverConfig | None' = None)

The transition must preserve the declared equilibrium-state shape even when the encoder, branch operators, constraints, solver, and readout are replaced. Test the transition by itself before testing the complete root solve.

Progressive Experiment Ladder

1. Equation and tensor contract

Objective: Make the state, conditioning variables, operator, and readout explicit.

Procedure:

• Write and evaluate the family equation: H_star=h_theta(H_star,G); U_hat=D(H_star); L_res=MSE(A U_hat-B)
• Declare every tensor axis, boundary, mask, graph, or physical unit.
• Check the transition output has exactly the same shape as the equilibrium state.

Acceptance checks:

• finite transition values
• shape-preserving state update
• all conditioning variables affect the intended branch

Evidence target: contract-verified.

2. Primitive mechanism reconstruction

Objective: Build the retained source mechanism from replaceable modules.

Procedure:

• encode-process-decode graph equilibrium
• separate interior incoming/outgoing and Neumann incoming messages
• fixed Dirichlet latent values, Broyden root solving, PDE residual, and Jacobian stabilization

Acceptance checks:

• primitive modules expose trainable parameters and gradients
• mechanism-specific invariance or constraint check passes
• direct transition evaluation is deterministic under a fixed seed

Evidence target: compact-verified.

3. Public abstraction equivalence

Objective: Verify that the assembled family evaluates the same transition as its primitives.

Procedure:

• Copy the primitive module parameters into the public family constructor.
• Evaluate one transition and one complete equilibrium with identical inputs.
• Compare outputs, residuals, and parameter gradients with declared tolerances.

Acceptance checks:

• transition outputs agree
• equilibrium residual is finite and decreases
• primitive and assembled gradients agree on the compact case

Evidence target: compact-verified.

4. Compact real or analytic task

Objective: Exercise training, evaluation, diagnostics, and serialization end to end.

Procedure:

• make_psi_poisson_grid gives a known mixed-boundary Poisson solution, directed graph, and residual matrix.

Acceptance checks:

• record finite-element residual
• record MSE against LU solution
• record boundary error
• record Jacobian norm
• checkpoint reload reproduces the recorded prediction
• result record contains data and configuration fingerprints

Evidence target: compact-verified.

5. Official-data subset

Objective: Validate the complete source data path before spending the full budget.

Procedure:

• Generate the 6000/2000/2000 mesh split with first-order elements, mixed boundaries, and approximately 500 training nodes per graph.
• Freeze preprocessing, split logic, metric code, and checkpoint format.
• Run a deterministic subset large enough to expose batching and memory failures.

Acceptance checks:

• dataset receipt and checksum are stored
• resume and evaluation paths reproduce the same subset metric
• memory and runtime are measured rather than estimated

Evidence target: subset-verified.

6. Source-scale reproduction or declared extension

Objective: Run the cited protocol, or change it explicitly as a SILVA extension.

Procedure:

• Generate the 6000/2000/2000 mesh split with first-order elements, mixed boundaries, and approximately 500 training nodes per graph.
• Convert each mesh to the SILVAPsiGNN tensor contract without densifying edges or finite-element matrices.
• Train residual, Jacobian, latent-consistency, and reconstruction terms, then evaluate new geometries, resolutions, boundaries, and initial states.
• paper mesh generator, GMSH first-order elements, 6000/2000/2000 split, and approximately 500 nodes
• finite-element residual matrices for training, mixed boundaries, optimizer groups, and seeds
• published residual, LU error, parameter count, and variable-resolution evaluation

Acceptance checks:

• all required artifacts are archived
• reported metrics use the cited evaluation protocol
• every architectural or training deviation is listed
• claims match the achieved evidence status

Evidence target: planned.

Data, Access, and Storage

Candidate datasets:

• paper synthetic unstructured Poisson meshes
• compact mixed-boundary finite-difference grids

Authoritative routes:

• https://arxiv.org/abs/2302.10891
• https://gmsh.info/

Access obligations:

• The benchmark is procedurally generated rather than a fixed public archive.
• Recreate first-order unstructured meshes and mixed boundaries from the paper protocol with Gmsh, then save generator parameters and mesh checksums.

Storage planning:

• Plan separately for node features, directed edge indices/features, targets, and optional sparse finite-element matrices.
• Measure one serialized mesh after preprocessing and multiply by 10,000; shard by graph count so matrices are never densified.

Preprocessing record:

• record dataset version, split, normalization, shape convention, and seed
• preserve masks, graph indices, boundaries, or physical units required by the domain

Metrics and Current Evidence

Required metrics:

• finite-element residual
• MSE against LU solution
• boundary error
• Jacobian norm
• root iterations

This family is verified through its listed mechanism tests and executed notebook. It is not included in a same-task comparison when another family does not share its input, state, output, and loss contract. The absence of a comparison row is therefore a scope decision, not missing implementation evidence.

Executed notebook paths:

• notebooks/package_api/29_silva_psi_gnn.ipynb

Mechanism tests:

• tests/test_emerging_equilibria.py

Compact Defaults

Option	Value
tier	'smoke'
config	SolverConfig(solver='broyden', max_iter=12, tol=1e-05, alpha=1.0, history=3, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=0, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=20, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Full Defaults

Option	Value
tier	'full'
config	SolverConfig(solver='broyden', max_iter=60, tol=1e-05, alpha=1.0, history=6, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=0, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=80, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Defaults establish a starting budget; the cited source protocol takes precedence whenever reproduction is the claim.

Source-Scale Checklist

• Generate the 6000/2000/2000 mesh split with first-order elements, mixed boundaries, and approximately 500 training nodes per graph.
• Convert each mesh to the SILVAPsiGNN tensor contract without densifying edges or finite-element matrices.
• Train residual, Jacobian, latent-consistency, and reconstruction terms, then evaluate new geometries, resolutions, boundaries, and initial states.

Benchmark-specific requirements:

• paper mesh generator, GMSH first-order elements, 6000/2000/2000 split, and approximately 500 nodes
• finite-element residual matrices for training, mixed boundaries, optimizer groups, and seeds
• published residual, LU error, parameter count, and variable-resolution evaluation

Required archived artifacts:

• machine-readable model and solver configuration
• dataset receipt with source revision, split, license, and checksum
• preprocessing and normalization record
• seeded training and evaluation log
• checkpoint and optimizer-resume state for trained experiments
• task metrics and equilibrium diagnostics in a machine-readable result
• runtime, peak-memory, device, precision, and dependency record
• declared deviations from the cited protocol

Reporting Rule

Report the achieved evidence status, not the intended one. A compact or subset run may validate the implementation and data path, but only a completed cited protocol supports a source-scale reproduction statement. Modified operators are valuable SILVA extensions when every deviation is named and measured.

Where to Go Next

Question	Page
Where are all family dossiers?	Family Dossier Index
How is a custom family assembled?	Advanced Extension Handbook
How are experiment stages represented in the API?	Research-Depth API
Which lab inspects every dossier?	Family Dossier Lab