SILVA Certified Equilibrium Lab¶
Derive signed interval propagation through a contractive equilibrium, test sampled perturbations, compute certified margins, and export the affine ReLU system [[97]] [[98]].
Numbered literature: [1], [4], [75], [91], [97], [98]. Each number opens the complete citation and its primary external source.
In [1]:
Copied!
from pathlib import Path
import sys
root = Path.cwd()
while root != root.parent and not (root / "src" / "silva_networks").exists():
root = root.parent
if not (root / "src" / "silva_networks").exists():
root = Path("/content/silva-networks")
sys.path.insert(0, str(root / "src"))
import matplotlib.pyplot as plt
import torch
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
torch.manual_seed(620)
from pathlib import Path
import sys
root = Path.cwd()
while root != root.parent and not (root / "src" / "silva_networks").exists():
root = root.parent
if not (root / "src" / "silva_networks").exists():
root = Path("/content/silva-networks")
sys.path.insert(0, str(root / "src"))
import matplotlib.pyplot as plt
import torch
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
torch.manual_seed(620)
Out[1]:
<torch._C.Generator at 0x122c30bb0>
1. Coupled Interval Fixed Point¶
For $A=A^++A^-$, $$\underline{Av}=A^+\underline v+A^-\overline v,\qquad \overline{Av}=A^+\overline v+A^-\underline v.$$
Applying these rules to $z=\phi(Wz+Ux+b)$ gives a joint lower/upper equilibrium. Contractivity prevents the interval recursion from diverging.
In [2]:
Copied!
from silva_networks import SILVACertifiedEquilibrium, SolverConfig
model = SILVACertifiedEquilibrium(
2, 16, 3, contraction=.7,
config=SolverConfig(solver="picard", max_iter=100, tol=1e-9,
backward_mode="unrolled", anderson_batch_dims=1))
points = torch.tensor([[-.7, -.3], [.2, .6], [.8, -.4], [-.2, .9]])
logits = model(points)
labels = logits.argmax(1)
bounds = model.interval_bounds(points-.08, points+.08)
certificate = model.certify(points, .08, labels)
print("labels:", labels.tolist())
print("margins:", certificate.margin.tolist())
print("certified:", certificate.certified.tolist())
from silva_networks import SILVACertifiedEquilibrium, SolverConfig
model = SILVACertifiedEquilibrium(
2, 16, 3, contraction=.7,
config=SolverConfig(solver="picard", max_iter=100, tol=1e-9,
backward_mode="unrolled", anderson_batch_dims=1))
points = torch.tensor([[-.7, -.3], [.2, .6], [.8, -.4], [-.2, .9]])
logits = model(points)
labels = logits.argmax(1)
bounds = model.interval_bounds(points-.08, points+.08)
certificate = model.certify(points, .08, labels)
print("labels:", labels.tolist())
print("margins:", certificate.margin.tolist())
print("certified:", certificate.certified.tolist())
labels: [0, 0, 0, 1] margins: [-0.08359481394290924, -0.04555016756057739, -0.049749746918678284, -0.05992189049720764] certified: [False, False, False, False]
2. Empirical Inclusion Check¶
Draw perturbations inside every input box. Their logits must remain between the certified lower and upper readout bounds. Sampling does not prove the certificate; it is an additional regression check.
In [3]:
Copied!
generator = torch.Generator().manual_seed(17)
samples=[]
for _ in range(100):
perturbation=(2*torch.rand(points.shape, generator=generator)-1)*.08
samples.append(model(points+perturbation).detach())
sampled=torch.stack(samples)
assert torch.all(sampled >= bounds.output_lower[None]-1e-6)
assert torch.all(sampled <= bounds.output_upper[None]+1e-6)
system=model.semialgebraic_system()
print("state row norm:", float(system.state_weight.abs().sum(-1).max()))
print("exported activation:", system.activation)
generator = torch.Generator().manual_seed(17)
samples=[]
for _ in range(100):
perturbation=(2*torch.rand(points.shape, generator=generator)-1)*.08
samples.append(model(points+perturbation).detach())
sampled=torch.stack(samples)
assert torch.all(sampled >= bounds.output_lower[None]-1e-6)
assert torch.all(sampled <= bounds.output_upper[None]+1e-6)
system=model.semialgebraic_system()
print("state row norm:", float(system.state_weight.abs().sum(-1).max()))
print("exported activation:", system.activation)
state row norm: 0.7000000476837158 exported activation: relu
In [4]:
Copied!
radii=torch.linspace(0, .2, 21); counts=[]; margins=[]
for radius in radii:
item=model.certify(points, radius, labels)
counts.append(int(item.certified.sum())); margins.append(float(item.margin.min()))
fig, axes=plt.subplots(1,2,figsize=(8,3.2),constrained_layout=True)
axes[0].step(radii,counts,where="post",color="#2563eb")
axes[0].set(xlabel="input radius",ylabel="certified examples",title="certificate curve")
axes[1].plot(radii,margins,color="#dc2626"); axes[1].axhline(0,color="#111827")
axes[1].set(xlabel="input radius",ylabel="minimum margin",title="worst margin")
plt.show()
radii=torch.linspace(0, .2, 21); counts=[]; margins=[]
for radius in radii:
item=model.certify(points, radius, labels)
counts.append(int(item.certified.sum())); margins.append(float(item.margin.min()))
fig, axes=plt.subplots(1,2,figsize=(8,3.2),constrained_layout=True)
axes[0].step(radii,counts,where="post",color="#2563eb")
axes[0].set(xlabel="input radius",ylabel="certified examples",title="certificate curve")
axes[1].plot(radii,margins,color="#dc2626"); axes[1].axhline(0,color="#111827")
axes[1].set(xlabel="input radius",ylabel="minimum margin",title="worst margin")
plt.show()
In [5]:
Copied!
from silva_networks import silva_family_experiment_protocol
for tier in silva_family_experiment_protocol("ibp_mondeq").tiers:
print(tier.tier, tier.dataset.name, tier.acceptance_checks[:2])
from silva_networks import silva_family_experiment_protocol
for tier in silva_family_experiment_protocol("ibp_mondeq").tiers:
print(tier.tier, tier.dataset.name, tier.acceptance_checks[:2])
smoke seeded convex or inverse fixture ('record natural accuracy', 'record certified accuracy by radius')
workstation MNIST or CIFAR-10 subset ('dataset receipt and checksum are stored', 'resume and evaluation paths reproduce the same subset metric')
full source optimization benchmark ('all required artifacts are archived', 'reported metrics use the cited evaluation protocol')
3. Source Route¶
Train and evaluate with the exact perturbation norm, radius, architecture, and certified-accuracy protocol from IBP-MonDEQ [[97]]. For semialgebraic certification, export the ReLU system and retain solver tolerances and external program settings [[98]].
From 58 Silva Certified Equilibrium 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 [6]:
Copied!
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¶
- 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.
In [7]:
Copied!
notebook_reproduction_record = {
"notebook": '58_silva_certified_equilibrium.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": '58_silva_certified_equilibrium.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[7]:
{'notebook': '58_silva_certified_equilibrium.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 transition 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 [8]:
Copied!
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('transition 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('transition 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)
transition 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 [9]:
Copied!
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('transition 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('transition 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 transition 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.