Implicit Functions and Automatic Differentiation¶
Let $z^\star$ solve
$$ F(z^\star,\theta)=f_\theta(z^\star,x)-z^\star=0. $$
Differentiate $F=0$:
$$ \frac{\partial f_\theta}{\partial z}\,dz +\frac{\partial f_\theta}{\partial \theta}\,d\theta -dz=0. $$
Collect the $dz$ terms:
$$ (I-J_f)\,dz=\frac{\partial f_\theta}{\partial \theta}\,d\theta. $$
Reverse-mode differentiation solves the transposed linear system
$$ (I-J_f^\top)u=g, $$
where $g=\partial \ell/\partial z^\star$.
Numbered literature: [3], [4], [13]. Each number opens the complete citation and its primary external source.
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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()
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import math
import numpy as np
import torch
import matplotlib.pyplot as plt
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
from silva_networks import resolve_device, SolverConfig
torch.manual_seed(7)
np.random.seed(7)
device = resolve_device("cuda" if torch.cuda.is_available() else "cpu")
device
import math
import numpy as np
import torch
import matplotlib.pyplot as plt
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
from silva_networks import resolve_device, SolverConfig
torch.manual_seed(7)
np.random.seed(7)
device = resolve_device("cuda" if torch.cuda.is_available() else "cpu")
device
Out[2]:
device(type='cpu')
Materialized Jacobian on a Small State¶
For a tiny state it is useful to materialize $J_f$. This makes the
matrix-free vjp and jvp calls easy to verify.
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from silva_networks import (
SILVAImplicitTransition,
fixed_point,
full_jacobian,
vjp,
jvp,
implicit_adjoint_solve,
)
transition = SILVAImplicitTransition(2, 2, spectral_scale=0.35).to(device)
x = torch.tensor([[0.3, -0.5]], device=device)
z0 = torch.zeros(1, 2, device=device)
def f(z):
return transition(z, x)
solve = fixed_point(f, z0, SolverConfig(max_iter=20, alpha=0.7))
J = full_jacobian(f, solve.z)
J
from silva_networks import (
SILVAImplicitTransition,
fixed_point,
full_jacobian,
vjp,
jvp,
implicit_adjoint_solve,
)
transition = SILVAImplicitTransition(2, 2, spectral_scale=0.35).to(device)
x = torch.tensor([[0.3, -0.5]], device=device)
z0 = torch.zeros(1, 2, device=device)
def f(z):
return transition(z, x)
solve = fixed_point(f, z0, SolverConfig(max_iter=20, alpha=0.7))
J = full_jacobian(f, solve.z)
J
Out[3]:
tensor([[-0.0380, -0.0973],
[ 0.1522, -0.2948]])
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probe = torch.randn_like(solve.z)
_, jvp_value = jvp(f, solve.z, probe)
vjp_value = vjp(f, solve.z, probe)
checks = {
"Jv_close": torch.allclose(J @ probe.reshape(-1), jvp_value.reshape(-1), atol=1e-5),
"JT_v_close": torch.allclose(J.T @ probe.reshape(-1), vjp_value.reshape(-1), atol=1e-5),
}
checks
probe = torch.randn_like(solve.z)
_, jvp_value = jvp(f, solve.z, probe)
vjp_value = vjp(f, solve.z, probe)
checks = {
"Jv_close": torch.allclose(J @ probe.reshape(-1), jvp_value.reshape(-1), atol=1e-5),
"JT_v_close": torch.allclose(J.T @ probe.reshape(-1), vjp_value.reshape(-1), atol=1e-5),
}
checks
Out[4]:
{'Jv_close': True, 'JT_v_close': True}
Adjoint Solve¶
The explicit matrix solution is
$$ u=(I-J_f^\top)^{-1}g. $$
The package helper computes the same object with VJP-backed GMRES.
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g = torch.tensor([[1.0, -0.25]], device=device)
explicit_u = torch.linalg.solve(torch.eye(2, device=device) - J.T, g.reshape(-1)).reshape_as(g)
gmres_u = implicit_adjoint_solve(f, solve.z, g, max_iter=8, tol=1e-7)
explicit_u.detach().cpu().numpy(), gmres_u.x.detach().cpu().numpy(), gmres_u.residuals
g = torch.tensor([[1.0, -0.25]], device=device)
explicit_u = torch.linalg.solve(torch.eye(2, device=device) - J.T, g.reshape(-1)).reshape_as(g)
gmres_u = implicit_adjoint_solve(f, solve.z, g, max_iter=8, tol=1e-7)
explicit_u.detach().cpu().numpy(), gmres_u.x.detach().cpu().numpy(), gmres_u.residuals
Out[5]:
(array([[ 0.9249382, -0.262553 ]], dtype=float32), array([[ 0.9249381, -0.262553 ]], dtype=float32), [1.0307763814926147, 0.03870135918259621, 1.5840151945667458e-07, 1.2013700256829907e-07])
Jacobian Spectrum¶
The spectral radius $\rho(J_f)$ is a local fixed-point diagnostic. A value below one is compatible with local contraction. A value near or above one means the solver may need damping, a different solver, or regularization.
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from silva_networks import spectral_radius, hutchinson_jacobian_norm
rho = spectral_radius(f, solve.z, iters=12)
fro = hutchinson_jacobian_norm(f, solve.z, samples=2, squared=False)
float(rho), float(fro.detach().cpu())
from silva_networks import spectral_radius, hutchinson_jacobian_norm
rho = spectral_radius(f, solve.z, iters=12)
fro = hutchinson_jacobian_norm(f, solve.z, samples=2, squared=False)
float(rho), float(fro.detach().cpu())
Out[6]:
(0.20792360603809357, 0.2742011845111847)
Citation and Sources¶
If this package or notebook 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 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
Background sources:
- Deep Implicit Layers tutorial: https://implicit-layers-tutorial.org/
- LocusLab DEQ repository: https://github.com/locuslab/deq
- Deep Equilibrium Models: https://arxiv.org/abs/1909.01377
- Multiscale Deep Equilibrium Models: https://arxiv.org/abs/2006.08656
- Stabilizing Equilibrium Models by Jacobian Regularization: https://arxiv.org/abs/2106.14342
The notebook is adapted to the silva_networks public API. It links to the
sources above and keep the examples package-native.
From 02 Implicit Autodiff 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 converged latent state z_star |
| Condition | differentiable parameters and external inputs |
| Repeated computation | the map whose Jacobian defines I - J_z f |
| Required invariants | agreement of dense, JVP, VJP, and matrix-free products |
| Replaceable components | transition, Jacobian product, linear solver, regularizer, and loss |
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.
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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 adjoint residual and gradient error against explicit differentiation. 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 dimension and matrix-free linear-solver iterations. 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¶
- Replace one component from this notebook while preserving its state and domain invariants.
- Write the replacement first as an independent reference function, then as a module, and assert one-step equivalence.
- Compare two solver configurations on the identical trained transition.
- Add a compact baseline and a predeclared metric threshold.
- Create a full-scale configuration without weakening the compact tests.
The complete authoring protocol is documented in Extending SILVA.
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notebook_reproduction_record = {
"notebook": '02_implicit_autodiff.ipynb',
"state": 'the converged latent state z_star',
"condition": 'differentiable parameters and external inputs',
"transition": 'the map whose Jacobian defines I - J_z f',
"invariants": 'agreement of dense, JVP, VJP, and matrix-free products',
"compact_metric": 'adjoint residual and gradient error against explicit differentiation',
"scale_axis": 'state dimension and matrix-free linear-solver iterations',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record
notebook_reproduction_record = {
"notebook": '02_implicit_autodiff.ipynb',
"state": 'the converged latent state z_star',
"condition": 'differentiable parameters and external inputs',
"transition": 'the map whose Jacobian defines I - J_z f',
"invariants": 'agreement of dense, JVP, VJP, and matrix-free products',
"compact_metric": 'adjoint residual and gradient error against explicit differentiation',
"scale_axis": 'state dimension and matrix-free linear-solver iterations',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record
Out[8]:
{'notebook': '02_implicit_autodiff.ipynb',
'state': 'the converged latent state z_star',
'condition': 'differentiable parameters and external inputs',
'transition': 'the map whose Jacobian defines I - J_z f',
'invariants': 'agreement of dense, JVP, VJP, and matrix-free products',
'compact_metric': 'adjoint residual and gradient error against explicit differentiation',
'scale_axis': 'state dimension and matrix-free linear-solver iterations'}
Worked Convergence and Sensitivity Study¶
The preceding example demonstrates one configured solve. This additional study changes the Jacobian 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 converged latent state z_star, and its repeated map, the map whose Jacobian defines I - J_z f. 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.
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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('Jacobian 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('Jacobian 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)
Jacobian 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
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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('Jacobian 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('Jacobian 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 Jacobian 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 | adjoint residual and gradient error against explicit differentiation |
| 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 | agreement of dense, JVP, VJP, and matrix-free products |
| Scale sweep | Change one of state dimension and matrix-free linear-solver iterations 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.