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SILVA Operator Options¶

The reference graph model keeps the Figure 1 decomposition explicit:

$$ z_{k+1} = (1-\alpha)z_k + \alpha\Phi\{S_\theta(x)+L_\theta(\tanh z_k,E)+G_\theta(\tanh z_k,b)\}. $$

hidden_dim, stack_alphas, graph_mode, attention_mode, solver settings, and readout dimensions are constructor arguments.

Numbered literature: [1], [15], [16], [17], [18], [19], [20], [29]. Each number opens the complete citation and its primary external source.

In [1]:

from pathlib import Path
import importlib.util
import subprocess
import sys

IN_COLAB = "google.colab" in sys.modules
REPO_URL = "https://github.com/jseluis/silva-networks.git"

def find_local_silva_root():
    candidates = [
        Path.cwd(),
        Path("/content/silva-networks"),
        Path("/content/drive/MyDrive/silva-networks"),
    ]
    root = Path.cwd()
    while root != root.parent:
        candidates.append(root)
        root = root.parent
    for candidate in candidates:
        if (candidate / "src" / "silva_networks").exists():
            return candidate
    return None

root = find_local_silva_root()
if root is not None:
    sys.path.insert(0, str(root / "src"))
elif IN_COLAB and importlib.util.find_spec("silva_networks") is None:
    subprocess.check_call([sys.executable, "-m", "pip", "install", f"git+{REPO_URL}"])
    root = Path.cwd()
else:
    root = Path.cwd()

from pathlib import Path import importlib.util import subprocess import sys IN_COLAB = "google.colab" in sys.modules REPO_URL = "https://github.com/jseluis/silva-networks.git" def find_local_silva_root(): candidates = [ Path.cwd(), Path("/content/silva-networks"), Path("/content/drive/MyDrive/silva-networks"), ] root = Path.cwd() while root != root.parent: candidates.append(root) root = root.parent for candidate in candidates: if (candidate / "src" / "silva_networks").exists(): return candidate return None root = find_local_silva_root() if root is not None: sys.path.insert(0, str(root / "src")) elif IN_COLAB and importlib.util.find_spec("silva_networks") is None: subprocess.check_call([sys.executable, "-m", "pip", "install", f"git+{REPO_URL}"]) root = Path.cwd() else: root = Path.cwd()

In [2]:

import torch
import matplotlib.pyplot as plt

plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
from silva_networks import (
    SILVAGraphPresetNetwork,
    SILVAMolecularRegressor,
    SILVAVisionVectorClassifier,
    SolverConfig,
    damped_spectral_radius,
    molecular_to_silva_graph,
    tabular_to_silva_graph,
)

torch.manual_seed(6)
x = torch.randn(9, 5)
y = (x[:, 0] > 0).long()
edge_index = torch.tensor(
    [[0, 1, 2, 3, 4, 5, 6, 7, 8],
     [1, 2, 3, 4, 5, 6, 7, 8, 0]],
    dtype=torch.long,
)

import torch import matplotlib.pyplot as plt plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300}) from silva_networks import ( SILVAGraphPresetNetwork, SILVAMolecularRegressor, SILVAVisionVectorClassifier, SolverConfig, damped_spectral_radius, molecular_to_silva_graph, tabular_to_silva_graph, ) torch.manual_seed(6) x = torch.randn(9, 5) y = (x[:, 0] > 0).long() edge_index = torch.tensor( [[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0]], dtype=torch.long, )

The SILVA ablations are available through operator names. Each row below uses the same input graph and changes only the local or global interaction. The ablation value is the string "none": it removes that branch from $L_\theta$ or $G_\theta$. Solver self-persistence $(1-\alpha)z_k$ is still present, so the no-local/no-global case is stimulus plus damped recurrent persistence.

In [3]:

cases = [
    ("full", "GAT", "simple"),
    ("local_only", "GAT", "none"),
    ("global_only", "none", "simple"),
    ("static_global", "GAT", "static"),
    ("topk_global", "GAT", "topk"),
]

case_rows = []
for name, graph_mode, attention_mode in cases:
    model = SILVAGraphPresetNetwork(
        in_dim=5,
        hidden_dim=[12, 8],
        out_dim=2,
        task="node",
        graph_mode=graph_mode,
        attention_mode=attention_mode,
        num_heads=2,
        k_neighbors=3,
        stack_alphas=[0.5, 0.2],
        max_iter=3,
    )
    result = model(x, edge_index=edge_index, return_results=True)
    residuals = [float(r.residual) for r in result.solver_results]
    case_rows.append({"case": name, "shape": tuple(result.output.shape), "residuals": residuals})
    print(name, tuple(result.output.shape), [round(value, 5) for value in residuals])
case_rows

cases = [ ("full", "GAT", "simple"), ("local_only", "GAT", "none"), ("global_only", "none", "simple"), ("static_global", "GAT", "static"), ("topk_global", "GAT", "topk"), ] case_rows = [] for name, graph_mode, attention_mode in cases: model = SILVAGraphPresetNetwork( in_dim=5, hidden_dim=[12, 8], out_dim=2, task="node", graph_mode=graph_mode, attention_mode=attention_mode, num_heads=2, k_neighbors=3, stack_alphas=[0.5, 0.2], max_iter=3, ) result = model(x, edge_index=edge_index, return_results=True) residuals = [float(r.residual) for r in result.solver_results] case_rows.append({"case": name, "shape": tuple(result.output.shape), "residuals": residuals}) print(name, tuple(result.output.shape), [round(value, 5) for value in residuals]) case_rows

full (9, 2) [5.69209, 5.83524]
local_only (9, 2) [4.29137, 5.87878]
global_only (9, 2) [2.68777, 5.45418]
static_global (9, 2) [4.39082, 6.33538]
topk_global (9, 2) [6.57354, 6.09027]

Out[3]:

[{'case': 'full',
  'shape': (9, 2),
  'residuals': [5.692088603973389, 5.835236072540283]},
 {'case': 'local_only',
  'shape': (9, 2),
  'residuals': [4.2913689613342285, 5.878783702850342]},
 {'case': 'global_only',
  'shape': (9, 2),
  'residuals': [2.6877706050872803, 5.454184055328369]},
 {'case': 'static_global',
  'shape': (9, 2),
  'residuals': [4.390822887420654, 6.335383892059326]},
 {'case': 'topk_global',
  'shape': (9, 2),
  'residuals': [6.573538780212402, 6.090270519256592]}]

The bar plot compares the largest final residual across the ablation stack. It is a quick numerical check that changing an operator still produces a finite solved state on the same input.

In [4]:

plt.figure(figsize=(6, 3))
plt.bar([row["case"] for row in case_rows], [max(row["residuals"]) for row in case_rows])
plt.xticks(rotation=30, ha="right")
plt.ylabel("max final residual")
plt.tight_layout()

plt.figure(figsize=(6, 3)) plt.bar([row["case"] for row in case_rows], [max(row["residuals"]) for row in case_rows]) plt.xticks(rotation=30, ha="right") plt.ylabel("max final residual") plt.tight_layout()

The generic API accepts per-layer solvers and custom branches. A custom branch only has to return a tensor shaped like the recurrent state.

In [5]:

class BiasLocal(torch.nn.Module):
    def __init__(self, dim):
        super().__init__()
        self.bias = torch.nn.Parameter(torch.zeros(dim))

    def forward(self, z, edge_index=None, edge_attr=None, batch=None):
        return torch.tanh(z + self.bias)

from silva_networks import SILVAGraphNetwork

custom = SILVAGraphNetwork(
    in_dim=5,
    hidden_dims=[12, 8, 8],
    out_dim=2,
    task="node",
    local=lambda dim, index: BiasLocal(dim),
    global_term=["mean", "simple", "topk_attention"],
    global_kwargs=[None, None, {"k": 4}],
    self_term=[None, "linear", None],
    config=[
        SolverConfig(solver="picard", max_iter=3, alpha=0.5),
        SolverConfig(solver="anderson", max_iter=3, alpha=0.35, history=3),
        SolverConfig(solver="broyden", max_iter=3, alpha=0.25),
    ],
)
custom(x, edge_index=edge_index).shape

class BiasLocal(torch.nn.Module): def __init__(self, dim): super().__init__() self.bias = torch.nn.Parameter(torch.zeros(dim)) def forward(self, z, edge_index=None, edge_attr=None, batch=None): return torch.tanh(z + self.bias) from silva_networks import SILVAGraphNetwork custom = SILVAGraphNetwork( in_dim=5, hidden_dims=[12, 8, 8], out_dim=2, task="node", local=lambda dim, index: BiasLocal(dim), global_term=["mean", "simple", "topk_attention"], global_kwargs=[None, None, {"k": 4}], self_term=[None, "linear", None], config=[ SolverConfig(solver="picard", max_iter=3, alpha=0.5), SolverConfig(solver="anderson", max_iter=3, alpha=0.35, history=3), SolverConfig(solver="broyden", max_iter=3, alpha=0.25), ], ) custom(x, edge_index=edge_index).shape

Out[5]:

torch.Size([9, 2])

Molecular SILVA accepts categorical atom/bond ids or continuous feature vectors. The continuous path uses explicit input projectors.

In [6]:

mol = molecular_to_silva_graph(
    x=torch.randn(5, 3),
    edge_index=torch.tensor([[0, 1, 2, 3, 4, 1], [1, 2, 0, 4, 3, 0]]),
    edge_attr=torch.randn(6, 2),
    batch=torch.tensor([0, 0, 0, 1, 1]),
    y=torch.tensor([0.3, -0.1]),
)
molecular_model = SILVAMolecularRegressor(
    hidden_dim=[9, 6],
    atom_feature_dim=3,
    bond_feature_dim=2,
    num_heads=3,
    alphas=(0.5, 0.2),
    max_iter=2,
    dropout=0.0,
    spectral_norm=False,
)
molecular_model(**mol.model_kwargs(), return_results=True).output

mol = molecular_to_silva_graph( x=torch.randn(5, 3), edge_index=torch.tensor([[0, 1, 2, 3, 4, 1], [1, 2, 0, 4, 3, 0]]), edge_attr=torch.randn(6, 2), batch=torch.tensor([0, 0, 0, 1, 1]), y=torch.tensor([0.3, -0.1]), ) molecular_model = SILVAMolecularRegressor( hidden_dim=[9, 6], atom_feature_dim=3, bond_feature_dim=2, num_heads=3, alphas=(0.5, 0.2), max_iter=2, dropout=0.0, spectral_norm=False, ) molecular_model(**mol.model_kwargs(), return_results=True).output

Out[6]:

tensor([-0.1630, -0.0136], grad_fn=<SqueezeBackward1>)

Lyapunov-style and Jacobian diagnostics can be run on any transition. Here the spectral-radius estimate is applied to the first SILVA layer transition.

In [7]:

layer = SILVAGraphPresetNetwork(
    in_dim=5,
    hidden_dim=8,
    out_dim=2,
    stack_alphas=[0.5],
    num_heads=2,
    max_iter=3,
).layers[0]
z0 = torch.zeros(x.shape[0], 8)
z_star = layer(x, edge_index=edge_index)
rho = damped_spectral_radius(lambda z: layer.f(z, x, edge_index=edge_index), z_star, alpha=0.5, iters=3)
rho

layer = SILVAGraphPresetNetwork( in_dim=5, hidden_dim=8, out_dim=2, stack_alphas=[0.5], num_heads=2, max_iter=3, ).layers[0] z0 = torch.zeros(x.shape[0], 8) z_star = layer(x, edge_index=edge_index) rho = damped_spectral_radius(lambda z: layer.f(z, x, edge_index=edge_index), z_star, alpha=0.5, iters=3) rho

Out[7]:

0.755165159702301

Citation and Sources¶

If this notebook or package is used, cite the software repository:

Dr. Jose Luis Silva. SILVA Networks. Version 1.2.2. MIT License.
https://github.com/jseluis/silva-networks
https://doi.org/10.5281/zenodo.21770098

When the work is connected to the SILVA Networks paper, cite the paper as well:

Jose Luis Lima de Jesus Silva. SILVA Networks as Structured Implicit Layers and
Vector Attractors via Dynamic Interaction Fields. 2026. arXiv:2607.28989.
https://arxiv.org/abs/2607.28989

Background references used in the tutorial suite include:

• Deep Equilibrium Models, Bai, Kolter, and Koltun, NeurIPS 2019: https://arxiv.org/abs/1909.01377
• Multiscale Deep Equilibrium Models, Bai, Koltun, and Kolter, NeurIPS 2020: https://arxiv.org/abs/2006.08656
• Stabilizing Equilibrium Models by Jacobian Regularization, Bai, Koltun, and Kolter, ICML 2021: https://arxiv.org/abs/2106.14342
• Graph Attention Networks, Velickovic et al., ICLR 2018: https://arxiv.org/abs/1710.10903
• Attention Is All You Need, Vaswani et al., 2017: https://arxiv.org/abs/1706.03762

From 06 Silva Operator Options to a Custom SILVA Family¶

The construction in this notebook can be separated into the universal conditioned-equilibrium contract

$$ z_0=I_\eta(x),\qquad z^\star=T_\theta(z^\star,x),\qquad \widehat y=Q_\psi(z^\star). $$

For this topic:

Part	Concrete interpretation
Equilibrium state	the tensor solved to equilibrium
Condition	the observed input or source tensor
Repeated computation	the state-preserving transition evaluated by the root solver
Required invariants	shape, device, dtype, finiteness, and differentiability
Replaceable components	initializer, source encoder, transition, readout, and solver

The initializer and source path are evaluated outside or alongside the root solve. Only the state-preserving transition is repeated. Replacing an internal architecture does not change this equation, provided the transition still maps the same state space into itself.

In [8]:

silva-extension-curriculum

import torch as silva_extension_torch
from torch import nn as silva_extension_nn

from silva_networks import (
    SILVAConditionedEquilibrium,
    SILVAZeroInitializer,
    SolverConfig,
    validate_silva_transition,
)


class NotebookExtensionTransition(silva_extension_nn.Module):
    def __init__(self, condition_dim=2, state_dim=3):
        super().__init__()
        self.source = silva_extension_nn.Linear(condition_dim, state_dim)
        self.state_field = silva_extension_nn.Sequential(
            silva_extension_nn.Linear(state_dim, 2 * state_dim),
            silva_extension_nn.Tanh(),
            silva_extension_nn.Linear(2 * state_dim, state_dim),
        )

    def forward(self, state, condition):
        return silva_extension_torch.tanh(
            self.source(condition) + 0.15 * self.state_field(state)
        )


silva_extension_torch.manual_seed(610)
notebook_condition = silva_extension_torch.linspace(-1.0, 1.0, 8).reshape(4, 2)
notebook_state0 = silva_extension_torch.zeros(4, 3)
notebook_transition = NotebookExtensionTransition()

notebook_report = validate_silva_transition(
    notebook_transition,
    notebook_state0,
    notebook_condition,
)
assert notebook_report.valid

with silva_extension_torch.no_grad():
    notebook_reference_step = silva_extension_torch.tanh(
        notebook_transition.source(notebook_condition)
        + 0.15 * notebook_transition.state_field(notebook_state0)
    )
silva_extension_torch.testing.assert_close(
    notebook_transition(notebook_state0, notebook_condition),
    notebook_reference_step,
)

notebook_custom_model = SILVAConditionedEquilibrium(
    notebook_transition,
    SILVAZeroInitializer(3),
    readout=silva_extension_nn.Linear(3, 1),
    config=SolverConfig(
        solver="picard",
        max_iter=40,
        tol=1e-7,
        backward_mode="implicit",
        backward_solver="gmres",
        anderson_batch_dims=1,
    ),
)
notebook_custom_result = notebook_custom_model(
    notebook_condition,
    return_result=True,
)
assert notebook_custom_result.output.shape == (4, 1)
assert notebook_custom_result.solver_result.residual < 1e-5

notebook_custom_result.output.square().mean().backward()
assert all(
    parameter.grad is not None and silva_extension_torch.isfinite(parameter.grad).all()
    for parameter in notebook_custom_model.parameters()
)
print("custom transition:", notebook_report)
print("equilibrium residual:", notebook_custom_result.solver_result.residual)

import torch as silva_extension_torch from torch import nn as silva_extension_nn from silva_networks import ( SILVAConditionedEquilibrium, SILVAZeroInitializer, SolverConfig, validate_silva_transition, ) class NotebookExtensionTransition(silva_extension_nn.Module): def __init__(self, condition_dim=2, state_dim=3): super().__init__() self.source = silva_extension_nn.Linear(condition_dim, state_dim) self.state_field = silva_extension_nn.Sequential( silva_extension_nn.Linear(state_dim, 2 * state_dim), silva_extension_nn.Tanh(), silva_extension_nn.Linear(2 * state_dim, state_dim), ) def forward(self, state, condition): return silva_extension_torch.tanh( self.source(condition) + 0.15 * self.state_field(state) ) silva_extension_torch.manual_seed(610) notebook_condition = silva_extension_torch.linspace(-1.0, 1.0, 8).reshape(4, 2) notebook_state0 = silva_extension_torch.zeros(4, 3) notebook_transition = NotebookExtensionTransition() notebook_report = validate_silva_transition( notebook_transition, notebook_state0, notebook_condition, ) assert notebook_report.valid with silva_extension_torch.no_grad(): notebook_reference_step = silva_extension_torch.tanh( notebook_transition.source(notebook_condition) + 0.15 * notebook_transition.state_field(notebook_state0) ) silva_extension_torch.testing.assert_close( notebook_transition(notebook_state0, notebook_condition), notebook_reference_step, ) notebook_custom_model = SILVAConditionedEquilibrium( notebook_transition, SILVAZeroInitializer(3), readout=silva_extension_nn.Linear(3, 1), config=SolverConfig( solver="picard", max_iter=40, tol=1e-7, backward_mode="implicit", backward_solver="gmres", anderson_batch_dims=1, ), ) notebook_custom_result = notebook_custom_model( notebook_condition, return_result=True, ) assert notebook_custom_result.output.shape == (4, 1) assert notebook_custom_result.solver_result.residual < 1e-5 notebook_custom_result.output.square().mean().backward() assert all( parameter.grad is not None and silva_extension_torch.isfinite(parameter.grad).all() for parameter in notebook_custom_model.parameters() ) print("custom transition:", notebook_report) print("equilibrium residual:", notebook_custom_result.solver_result.residual)

custom transition: SILVATransitionReport(state_shape=(4, 3), output_shape=(4, 3), preserves_shape=True, preserves_device=True, preserves_dtype=True, finite=True, differentiable=True, parameter_count=54)
equilibrium residual: 5.960464477539063e-08

Numerical Equivalence, Compact Reproduction, and Scale¶

Before training, compare one packaged transition with an independently written update:

$$ e_{\mathrm{step}} =\frac{\|T_\theta(z,x)-T_{\mathrm{ref}}(z,x)\|_2} {\|T_{\mathrm{ref}}(z,x)\|_2+\varepsilon}. $$

After solving, report the fixed-point residual separately:

$$ e_{\mathrm{fp}} =\frac{\|T_\theta(z^\star,x)-z^\star\|_2} {\|z^\star\|_2+\varepsilon}. $$

For this notebook, a compact reproduction must declare and assert fixed-point residual and task error against a deterministic target. A full experiment must additionally record the source dataset version and split, preprocessing, architecture widths, solver and optimizer schedules, random seeds, baseline configuration, checkpoints, and every deviation from the cited protocol.

The principal scaling axes are state width, batch size, and data volume. Increase one axis at a time, retain the compact deterministic case as a regression test, and record task error, domain-specific residual, forward residual, backward linear residual, memory use, and runtime independently.

Extension Exercises¶

1. Replace one component from this notebook while preserving its state and domain invariants.
2. Write the replacement first as an independent reference function, then as a module, and assert one-step equivalence.
3. Compare two solver configurations on the identical trained transition.
4. Add a compact baseline and a predeclared metric threshold.
5. Create a full-scale configuration without weakening the compact tests.

The complete authoring protocol is documented in Extending SILVA.

In [9]:

silva-extension-curriculum

notebook_reproduction_record = {
    "notebook": '06_silva_operator_options.ipynb',
    "state": 'the tensor solved to equilibrium',
    "condition": 'the observed input or source tensor',
    "transition": 'the state-preserving transition evaluated by the root solver',
    "invariants": 'shape, device, dtype, finiteness, and differentiability',
    "compact_metric": 'fixed-point residual and task error against a deterministic target',
    "scale_axis": 'state width, batch size, and data volume',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record

notebook_reproduction_record = { "notebook": '06_silva_operator_options.ipynb', "state": 'the tensor solved to equilibrium', "condition": 'the observed input or source tensor', "transition": 'the state-preserving transition evaluated by the root solver', "invariants": 'shape, device, dtype, finiteness, and differentiability', "compact_metric": 'fixed-point residual and task error against a deterministic target', "scale_axis": 'state width, batch size, and data volume', } assert all(notebook_reproduction_record.values()) notebook_reproduction_record

Out[9]:

{'notebook': '06_silva_operator_options.ipynb',
 'state': 'the tensor solved to equilibrium',
 'condition': 'the observed input or source tensor',
 'transition': 'the state-preserving transition evaluated by the root solver',
 'invariants': 'shape, device, dtype, finiteness, and differentiability',
 'compact_metric': 'fixed-point residual and task error against a deterministic target',
 'scale_axis': 'state width, batch size, and data volume'}

Worked Convergence and Sensitivity Study¶

The preceding example demonstrates one configured solve. This additional study changes the operator feedback factor while keeping the source fixed, so solver effort and implicit sensitivity can be read separately from task behavior. Locally, one eigendirection of a nonlinear transition can be represented by

$$ z_{k+1} = \rho z_k + u, \qquad 0 \leq \rho < 1. $$

Its equilibrium is

$$ z^\star = \frac{u}{1-\rho}. $$

Subtracting the fixed-point equation from the iteration gives the exact error recursion

$$ e_{k+1} = \rho e_k, \qquad |e_k| = \rho^k |e_0|. $$

For a requested absolute tolerance $\tau$, the idealized iteration estimate is

$$ k \geq \frac{\log(\tau/|e_0|)}{\log \rho}. $$

The same factor controls sensitivity. Differentiating the equilibrium with respect to the source gives

$$ \frac{\partial z^\star}{\partial u} =\frac{1}{1-\rho}. $$

Thus a transition can remain contractive while becoming expensive and highly sensitive as $\rho$ approaches one. The table and figure below measure this effect rather than merely stating it. They provide a reference envelope for the notebook's actual state, the tensor solved to equilibrium, and its repeated map, the state-preserving transition evaluated by the root solver. The scalar study does not replace the domain model; it supplies a result whose convergence rate and derivative are known exactly, so the same reporting code can be trusted before it is applied to the larger transition.

In [10]:

silva-extension-curriculum

import math as silva_deepening_math
import torch as silva_deepening_torch

silva_deepening_rates = (0.20, 0.45, 0.70, 0.85)
silva_deepening_source = 0.35
silva_deepening_tolerance = 1e-8
silva_deepening_histories = {}
silva_deepening_rows = []

for silva_deepening_rho in silva_deepening_rates:
    silva_deepening_state = silva_deepening_torch.tensor(0.0)
    silva_deepening_exact = silva_deepening_source / (1.0 - silva_deepening_rho)
    silva_deepening_history = []
    for silva_deepening_iteration in range(1, 241):
        silva_deepening_next = (
            silva_deepening_rho * silva_deepening_state + silva_deepening_source
        )
        silva_deepening_residual = abs(
            float(silva_deepening_next - silva_deepening_state)
        )
        silva_deepening_history.append(silva_deepening_residual)
        silva_deepening_state = silva_deepening_next
        if silva_deepening_residual < silva_deepening_tolerance:
            break

    silva_deepening_u = silva_deepening_torch.tensor(
        silva_deepening_source, requires_grad=True
    )
    silva_deepening_solution = silva_deepening_u / (1.0 - silva_deepening_rho)
    silva_deepening_solution.backward()
    silva_deepening_expected_sensitivity = 1.0 / (1.0 - silva_deepening_rho)
    silva_deepening_gradient_error = abs(
        float(silva_deepening_u.grad) - silva_deepening_expected_sensitivity
    )
    silva_deepening_histories[silva_deepening_rho] = silva_deepening_history
    silva_deepening_rows.append(
        (
            silva_deepening_rho,
            silva_deepening_iteration,
            silva_deepening_history[-1],
            abs(float(silva_deepening_state) - silva_deepening_exact),
            float(silva_deepening_u.grad),
            silva_deepening_gradient_error,
        )
    )

print('operator feedback factor')
print("rho | iterations | final residual | exact-state error | sensitivity | gradient error")
for silva_deepening_row in silva_deepening_rows:
    print(
        f"{silva_deepening_row[0]:.2f} | {silva_deepening_row[1]:3d} | "
        f"{silva_deepening_row[2]:.3e} | {silva_deepening_row[3]:.3e} | "
        f"{silva_deepening_row[4]:.4f} | {silva_deepening_row[5]:.3e}"
    )

assert all(row[2] < silva_deepening_tolerance for row in silva_deepening_rows)
assert all(row[3] < 1e-6 for row in silva_deepening_rows)
assert all(row[5] < 1e-6 for row in silva_deepening_rows)

import math as silva_deepening_math import torch as silva_deepening_torch silva_deepening_rates = (0.20, 0.45, 0.70, 0.85) silva_deepening_source = 0.35 silva_deepening_tolerance = 1e-8 silva_deepening_histories = {} silva_deepening_rows = [] for silva_deepening_rho in silva_deepening_rates: silva_deepening_state = silva_deepening_torch.tensor(0.0) silva_deepening_exact = silva_deepening_source / (1.0 - silva_deepening_rho) silva_deepening_history = [] for silva_deepening_iteration in range(1, 241): silva_deepening_next = ( silva_deepening_rho * silva_deepening_state + silva_deepening_source ) silva_deepening_residual = abs( float(silva_deepening_next - silva_deepening_state) ) silva_deepening_history.append(silva_deepening_residual) silva_deepening_state = silva_deepening_next if silva_deepening_residual < silva_deepening_tolerance: break silva_deepening_u = silva_deepening_torch.tensor( silva_deepening_source, requires_grad=True ) silva_deepening_solution = silva_deepening_u / (1.0 - silva_deepening_rho) silva_deepening_solution.backward() silva_deepening_expected_sensitivity = 1.0 / (1.0 - silva_deepening_rho) silva_deepening_gradient_error = abs( float(silva_deepening_u.grad) - silva_deepening_expected_sensitivity ) silva_deepening_histories[silva_deepening_rho] = silva_deepening_history silva_deepening_rows.append( ( silva_deepening_rho, silva_deepening_iteration, silva_deepening_history[-1], abs(float(silva_deepening_state) - silva_deepening_exact), float(silva_deepening_u.grad), silva_deepening_gradient_error, ) ) print('operator feedback factor') print("rho | iterations | final residual | exact-state error | sensitivity | gradient error") for silva_deepening_row in silva_deepening_rows: print( f"{silva_deepening_row[0]:.2f} | {silva_deepening_row[1]:3d} | " f"{silva_deepening_row[2]:.3e} | {silva_deepening_row[3]:.3e} | " f"{silva_deepening_row[4]:.4f} | {silva_deepening_row[5]:.3e}" ) assert all(row[2] < silva_deepening_tolerance for row in silva_deepening_rows) assert all(row[3] < 1e-6 for row in silva_deepening_rows) assert all(row[5] < 1e-6 for row in silva_deepening_rows)

operator feedback factor
rho | iterations | final residual | exact-state error | sensitivity | gradient error
0.20 |  12 | 0.000e+00 | 5.551e-17 | 1.2500 | 0.000e+00
0.45 |  23 | 0.000e+00 | 1.084e-08 | 1.8182 | 6.502e-08
0.70 |  45 | 0.000e+00 | 1.589e-07 | 3.3333 | 7.947e-08
0.85 |  93 | 0.000e+00 | 5.563e-07 | 6.6667 | 1.589e-07

In [11]:

silva-extension-curriculum

import matplotlib.pyplot as silva_deepening_plt

silva_deepening_plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
silva_deepening_figure, silva_deepening_axes = silva_deepening_plt.subplots(
    1, 2, figsize=(8.6, 3.2)
)
for silva_deepening_rho, silva_deepening_history in silva_deepening_histories.items():
    silva_deepening_axes[0].semilogy(
        range(1, len(silva_deepening_history) + 1),
        silva_deepening_history,
        marker="o",
        markersize=2,
        linewidth=1.2,
        label=f"rho={silva_deepening_rho:.2f}",
    )
silva_deepening_axes[0].axhline(
    silva_deepening_tolerance, color="black", linestyle="--", linewidth=0.9
)
silva_deepening_axes[0].set_xlabel("iteration")
silva_deepening_axes[0].set_ylabel("absolute residual")
silva_deepening_axes[0].set_title("Residual trajectories")
silva_deepening_axes[0].legend(fontsize=7)

silva_deepening_axes[1].plot(
    [row[0] for row in silva_deepening_rows],
    [row[1] for row in silva_deepening_rows],
    marker="o",
    label="iterations",
)
silva_deepening_sensitivity_axis = silva_deepening_axes[1].twinx()
silva_deepening_sensitivity_axis.plot(
    [row[0] for row in silva_deepening_rows],
    [row[4] for row in silva_deepening_rows],
    color="tab:red",
    marker="s",
    label="sensitivity",
)
silva_deepening_axes[1].set_xlabel('operator feedback factor')
silva_deepening_axes[1].set_ylabel("iterations")
silva_deepening_sensitivity_axis.set_ylabel("implicit sensitivity", color="tab:red")
silva_deepening_axes[1].set_title("Cost and sensitivity")
silva_deepening_figure.tight_layout()
silva_deepening_plt.show()

import matplotlib.pyplot as silva_deepening_plt silva_deepening_plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300}) silva_deepening_figure, silva_deepening_axes = silva_deepening_plt.subplots( 1, 2, figsize=(8.6, 3.2) ) for silva_deepening_rho, silva_deepening_history in silva_deepening_histories.items(): silva_deepening_axes[0].semilogy( range(1, len(silva_deepening_history) + 1), silva_deepening_history, marker="o", markersize=2, linewidth=1.2, label=f"rho={silva_deepening_rho:.2f}", ) silva_deepening_axes[0].axhline( silva_deepening_tolerance, color="black", linestyle="--", linewidth=0.9 ) silva_deepening_axes[0].set_xlabel("iteration") silva_deepening_axes[0].set_ylabel("absolute residual") silva_deepening_axes[0].set_title("Residual trajectories") silva_deepening_axes[0].legend(fontsize=7) silva_deepening_axes[1].plot( [row[0] for row in silva_deepening_rows], [row[1] for row in silva_deepening_rows], marker="o", label="iterations", ) silva_deepening_sensitivity_axis = silva_deepening_axes[1].twinx() silva_deepening_sensitivity_axis.plot( [row[0] for row in silva_deepening_rows], [row[4] for row in silva_deepening_rows], color="tab:red", marker="s", label="sensitivity", ) silva_deepening_axes[1].set_xlabel('operator feedback factor') silva_deepening_axes[1].set_ylabel("iterations") silva_deepening_sensitivity_axis.set_ylabel("implicit sensitivity", color="tab:red") silva_deepening_axes[1].set_title("Cost and sensitivity") silva_deepening_figure.tight_layout() silva_deepening_plt.show()

Reading and Extending the Result¶

The measured residual curves become flatter as the operator feedback factor increases. The iteration count and the exact sensitivity rise together, but they answer different questions: iterations measure numerical work, while sensitivity describes how strongly the equilibrium reacts to the source. The gradient-error column verifies the differentiation path against the analytic derivative.

Apply the same separation to this notebook's full model:

Report	Notebook-specific interpretation
Task evidence	fixed-point residual and task error against a deterministic target
Forward residual	Re-evaluate the complete transition at the returned state
Empirical rate	Compare consecutive residuals only after the transient regime
Backward residual	Record the linear-adjoint stopping value independently
Sensitivity	Perturb one declared source field while preserving all other inputs
Structural checks	shape, device, dtype, finiteness, and differentiability
Scale sweep	Change one of state width, batch size, and data volume at a time

A richer experiment should now repeat the sweep with at least two forward solvers, two tolerances, and multiple seeds. Keep model parameters and data identical when comparing solvers. Then change one architecture or data-scale axis, retain the compact analytic study as a regression test, and report task quality, residuals, iterations, runtime, memory, gradient norms, and failed convergence cases together.

Where to Go Next¶

Question	Page
How does each named branch affect the transition?	SILVA Operators
Which larger mappings can occupy one point?	Point Architecture Catalog
Where are all operators combined in one example?	Full Cortex Operators