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Equation-to-Code Walkthrough¶

This notebook follows one thread from the full SILVA equation to executable package objects:

$$ z^\star = \Psi_\theta\!\left( S_\theta(x) +H_\theta(a(z^\star)) +L_\theta(a(z^\star),E) +G_\theta(a(z^\star),b) \right). $$

By setting terms to zero or choosing different operators, the same package API recovers compact DEQ layers, message-passing DEQs, and full SILVA graph layers.

Numbered literature: [1], [4], [10], [11], [13]. 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 (
    SILVAGraphNetwork,
    SolverConfig,
    full_jacobian,
    make_knn_edge_index,
    resolve_device,
    silva_deq_reduction_layer,
    silva_message_passing_reduction_layer,
    stability_report,
)

torch.manual_seed(8)
device = resolve_device("cuda" if torch.cuda.is_available() else "cpu")
device

import torch import matplotlib.pyplot as plt plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300}) from silva_networks import ( SILVAGraphNetwork, SolverConfig, full_jacobian, make_knn_edge_index, resolve_device, silva_deq_reduction_layer, silva_message_passing_reduction_layer, stability_report, ) torch.manual_seed(8) device = resolve_device("cuda" if torch.cuda.is_available() else "cpu") device

Out[2]:

device(type='cpu')

Step 1: Compact DEQ as a SILVA Reduction¶

Start from the full equation and set

$$ L_\theta=0,\qquad G_\theta=0,\qquad a(z)=z,\qquad H_\theta(z)=W_z z. $$

With $S_\theta(x)=W_xx+b$ and $\Psi=\tanh$, substitution gives

$$ z^\star = \tanh(W_xx+b+W_z z^\star). $$

In [3]:

x = torch.randn(12, 4, device=device)
deq_layer = silva_deq_reduction_layer(
    in_dim=4,
    hidden_dim=6,
    config=SolverConfig(solver="anderson", max_iter=12, alpha=0.6, history=4),
).to(device)

z = torch.randn(12, 6, device=device)
manual = torch.tanh(deq_layer.stimulus(x) + deq_layer.self_term(z))
from_layer = deq_layer.f(z, x)
float((manual - from_layer).abs().max().detach().cpu())

x = torch.randn(12, 4, device=device) deq_layer = silva_deq_reduction_layer( in_dim=4, hidden_dim=6, config=SolverConfig(solver="anderson", max_iter=12, alpha=0.6, history=4), ).to(device) z = torch.randn(12, 6, device=device) manual = torch.tanh(deq_layer.stimulus(x) + deq_layer.self_term(z)) from_layer = deq_layer.f(z, x) float((manual - from_layer).abs().max().detach().cpu())

Out[3]:

0.0

In [4]:

deq_result = deq_layer(x, return_result=True)
deq_result.z.shape, deq_result.iterations, deq_result.residual

deq_result = deq_layer(x, return_result=True) deq_result.z.shape, deq_result.iterations, deq_result.residual

Out[4]:

(torch.Size([12, 6]), 12, 0.0026819880586117506)

Step 2: Message Passing as a Local SILVA Field¶

Keep a local operator and remove the learned self and global branches:

$$ z^\star=\Psi_\theta(S_\theta(x)+L_\theta(a(z^\star),E)). $$

For a graph, $E$ is encoded by edge_index. The first row stores sources and the second row stores destinations.

In [5]:

edge_index = make_knn_edge_index(x, k=3, undirected=True, device=device)
mp_layer = silva_message_passing_reduction_layer(
    in_dim=4,
    hidden_dim=6,
    local="gat",
    local_kwargs={"heads": 2},
    config=SolverConfig(solver="picard", max_iter=8, alpha=0.5),
).to(device)

mp_result = mp_layer(x, edge_index=edge_index, return_result=True)
mp_result.z.shape, edge_index.shape, mp_result.residual

edge_index = make_knn_edge_index(x, k=3, undirected=True, device=device) mp_layer = silva_message_passing_reduction_layer( in_dim=4, hidden_dim=6, local="gat", local_kwargs={"heads": 2}, config=SolverConfig(solver="picard", max_iter=8, alpha=0.5), ).to(device) mp_result = mp_layer(x, edge_index=edge_index, return_result=True) mp_result.z.shape, edge_index.shape, mp_result.residual

Out[5]:

(torch.Size([12, 6]), torch.Size([2, 50]), 1.6806175708770752)

Step 3: Turn the Full SILVA Field Back On¶

A stack can use different branch choices and solver settings per layer:

$$ z_{\ell}^{\star} = \Psi_{\theta_\ell} \left( S_{\theta_\ell}(x_\ell) +H_{\theta_\ell} +L_{\theta_\ell} +G_{\theta_\ell} \right). $$

In [6]:

model = SILVAGraphNetwork(
    in_dim=4,
    hidden_dims=[10, 8],
    out_dim=3,
    task="node",
    local=["graph", "gat"],
    global_term=["mean", "topk"],
    self_term=["none", "linear"],
    local_kwargs=[None, {"heads": 2}],
    global_kwargs=[None, {"k": 4}],
    config=[
        SolverConfig(solver="picard", max_iter=6, alpha=0.5),
        SolverConfig(solver="anderson", max_iter=8, alpha=0.35, history=4),
    ],
).to(device)

out = model(x, edge_index=edge_index, return_results=True)
out.output.shape, out.state.shape, [r.solver for r in out.solver_results]

model = SILVAGraphNetwork( in_dim=4, hidden_dims=[10, 8], out_dim=3, task="node", local=["graph", "gat"], global_term=["mean", "topk"], self_term=["none", "linear"], local_kwargs=[None, {"heads": 2}], global_kwargs=[None, {"k": 4}], config=[ SolverConfig(solver="picard", max_iter=6, alpha=0.5), SolverConfig(solver="anderson", max_iter=8, alpha=0.35, history=4), ], ).to(device) out = model(x, edge_index=edge_index, return_results=True) out.output.shape, out.state.shape, [r.solver for r in out.solver_results]

Out[6]:

(torch.Size([12, 3]), torch.Size([12, 8]), ['picard', 'anderson'])

Step 4: Diagnose the Equilibrium¶

The residual is

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

For small states, the full Jacobian can be materialized. For larger states, matrix-free vjp and jvp calls are preferred.

In [7]:

small_x = x[:2]
small_layer = silva_deq_reduction_layer(
    in_dim=4,
    hidden_dim=3,
    config=SolverConfig(solver="picard", max_iter=10, alpha=0.5),
).to(device)
small_result = small_layer(small_x, return_result=True)

J = full_jacobian(lambda state: small_layer.f(state, small_x), small_result.z)
report = stability_report(
    lambda state: small_layer.f(state, small_x),
    small_result.z,
    samples=2,
    iters=4,
)
J.shape, report

small_x = x[:2] small_layer = silva_deq_reduction_layer( in_dim=4, hidden_dim=3, config=SolverConfig(solver="picard", max_iter=10, alpha=0.5), ).to(device) small_result = small_layer(small_x, return_result=True) J = full_jacobian(lambda state: small_layer.f(state, small_x), small_result.z) report = stability_report( lambda state: small_layer.f(state, small_x), small_result.z, samples=2, iters=4, ) J.shape, report

Out[7]:

(torch.Size([6, 6]),
 StabilityReport(residual=0.004042531829327345, spectral_radius=0.3276681900024414, jacobian_norm_estimate=0.6150436997413635, samples=2))

In [8]:

plt.figure(figsize=(5, 3))
plt.plot(deq_result.residuals, marker="o", label="DEQ reduction")
plt.plot(mp_result.residuals, marker="o", label="message-passing reduction")
for index, solver_result in enumerate(out.solver_results):
    plt.plot(solver_result.residuals, marker="o", label=f"full SILVA layer {index}")
plt.yscale("log")
plt.xlabel("solver step")
plt.ylabel("residual")
plt.legend()
plt.tight_layout()

plt.figure(figsize=(5, 3)) plt.plot(deq_result.residuals, marker="o", label="DEQ reduction") plt.plot(mp_result.residuals, marker="o", label="message-passing reduction") for index, solver_result in enumerate(out.solver_results): plt.plot(solver_result.residuals, marker="o", label=f"full SILVA layer {index}") plt.yscale("log") plt.xlabel("solver step") plt.ylabel("residual") plt.legend() plt.tight_layout()

Citation¶

If this notebook or package is used, cite:

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 methodology, cite the SILVA Networks 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

Related sources:

• Deep Equilibrium Models: https://arxiv.org/abs/1909.01377
• Graph Attention Networks: https://arxiv.org/abs/1710.10903
• TorchDEQ: https://github.com/locuslab/torchdeq

From 08 Equation To Code Walkthrough 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 evolving or terminal physical state
Condition	time, initial condition, and external forcing
Repeated computation	an explicit flow step or residual field T(z, x) - z
Required invariants	time-step shape, initial condition, and integration consistency
Replaceable components	vector field, integrator, equilibrium transition, readout, and tolerances

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 [9]:

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 trajectory error, terminal fixed-point residual, and conservation error. 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 time horizon, step count, state dimension, and stiffness. 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 [10]:

silva-extension-curriculum

notebook_reproduction_record = {
    "notebook": '08_equation_to_code_walkthrough.ipynb',
    "state": 'the evolving or terminal physical state',
    "condition": 'time, initial condition, and external forcing',
    "transition": 'an explicit flow step or residual field T(z, x) - z',
    "invariants": 'time-step shape, initial condition, and integration consistency',
    "compact_metric": 'trajectory error, terminal fixed-point residual, and conservation error',
    "scale_axis": 'time horizon, step count, state dimension, and stiffness',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record

notebook_reproduction_record = { "notebook": '08_equation_to_code_walkthrough.ipynb', "state": 'the evolving or terminal physical state', "condition": 'time, initial condition, and external forcing', "transition": 'an explicit flow step or residual field T(z, x) - z', "invariants": 'time-step shape, initial condition, and integration consistency', "compact_metric": 'trajectory error, terminal fixed-point residual, and conservation error', "scale_axis": 'time horizon, step count, state dimension, and stiffness', } assert all(notebook_reproduction_record.values()) notebook_reproduction_record

Out[10]:

{'notebook': '08_equation_to_code_walkthrough.ipynb',
 'state': 'the evolving or terminal physical state',
 'condition': 'time, initial condition, and external forcing',
 'transition': 'an explicit flow step or residual field T(z, x) - z',
 'invariants': 'time-step shape, initial condition, and integration consistency',
 'compact_metric': 'trajectory error, terminal fixed-point residual, and conservation error',
 'scale_axis': 'time horizon, step count, state dimension, and stiffness'}

Worked Convergence and Sensitivity Study¶

The preceding example demonstrates one configured solve. This additional study changes the implicit dynamics 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 evolving or terminal physical state, and its repeated map, an explicit flow step or residual field T(z, x) - z. 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 [11]:

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('implicit dynamics 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('implicit dynamics 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)

implicit dynamics 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 [12]:

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('implicit dynamics 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('implicit dynamics 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 implicit dynamics 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	trajectory error, terminal fixed-point residual, and conservation error
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	time-step shape, initial condition, and integration consistency
Scale sweep	Change one of time horizon, step count, state dimension, and stiffness 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
Where is the complete derivation sequence explained?	Derivation Workbook
How is a full SILVA layer assembled?	SILVA From Scratch
Which layer contracts implement the derived branches?	Layers API