SILVA Monotone Operator Equilibrium¶
This lab derives the monotone inclusion, runs forward-backward and Peaceman-Rachford splitting on the same known solution, checks the certificate, and shows how a custom structured operator enters SILVA. The defining mechanism follows monDEQ [75].
Numbered literature: [1], [4], [75], [81]. 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
REPO_URL = "https://github.com/jseluis/silva-networks.git"
def find_local_silva_root():
candidates = [Path.cwd(), Path("/content/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 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
REPO_URL = "https://github.com/jseluis/silva-networks.git"
def find_local_silva_root():
candidates = [Path.cwd(), Path("/content/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 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 matplotlib.pyplot as plt
import torch
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
torch.manual_seed(91)
from torch import nn
from silva_networks import (
SILVAMonotoneOperatorEquilibrium,
SolverConfig,
make_monotone_operator_dataset,
)
import matplotlib.pyplot as plt
import torch
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
torch.manual_seed(91)
from torch import nn
from silva_networks import (
SILVAMonotoneOperatorEquilibrium,
SolverConfig,
make_monotone_operator_dataset,
)
1. From Fixed Point to Monotone Inclusion¶
$$ z^\star=\operatorname{prox}_f(Wz^\star+Ux+b) $$
is equivalent to
$$ 0\in(I-W)z^\star-Ux-b+\partial f(z^\star). $$
The source parameterization
$$ W=(1-m)I-A^\mathsf{T}A+B-B^\mathsf{T} $$
gives
$$ \operatorname{Sym}(I-W)=mI+A^\mathsf{T}A\succeq mI. $$
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data = make_monotone_operator_dataset(samples=16, seed=91)
class KnownMonotoneOperator(nn.Module):
def __init__(self, matrix):
super().__init__()
self.register_buffer("weight", matrix)
def forward(self, state):
return state @ self.weight.T
def resolvent(self, values, step_size):
identity = torch.eye(self.weight.shape[0], device=values.device)
system = (1 + step_size) * identity - step_size * self.weight
return torch.linalg.solve(system, values.T).T
def monotonicity_certificate(self):
identity = torch.eye(self.weight.shape[0], device=self.weight.device)
symmetric = identity - 0.5 * (self.weight + self.weight.T)
return torch.linalg.eigvalsh(symmetric).min()
source = nn.Linear(4, 6)
readout = nn.Linear(6, 2, bias=False)
with torch.no_grad():
source.weight.copy_(data.source)
source.bias.copy_(data.bias)
readout.weight.copy_(data.readout)
operator = KnownMonotoneOperator(data.recurrent)
print("certificate:", float(operator.monotonicity_certificate()))
print("known state shape:", data.equilibrium.shape)
data = make_monotone_operator_dataset(samples=16, seed=91)
class KnownMonotoneOperator(nn.Module):
def __init__(self, matrix):
super().__init__()
self.register_buffer("weight", matrix)
def forward(self, state):
return state @ self.weight.T
def resolvent(self, values, step_size):
identity = torch.eye(self.weight.shape[0], device=values.device)
system = (1 + step_size) * identity - step_size * self.weight
return torch.linalg.solve(system, values.T).T
def monotonicity_certificate(self):
identity = torch.eye(self.weight.shape[0], device=self.weight.device)
symmetric = identity - 0.5 * (self.weight + self.weight.T)
return torch.linalg.eigvalsh(symmetric).min()
source = nn.Linear(4, 6)
readout = nn.Linear(6, 2, bias=False)
with torch.no_grad():
source.weight.copy_(data.source)
source.bias.copy_(data.bias)
readout.weight.copy_(data.readout)
operator = KnownMonotoneOperator(data.recurrent)
print("certificate:", float(operator.monotonicity_certificate()))
print("known state shape:", data.equilibrium.shape)
certificate: 0.75 known state shape: torch.Size([16, 6])
2. Two Splittings, One Equilibrium¶
Forward-backward uses
$$ z_{k+1}=\operatorname{prox}_{af} \left((1-a)z_k+a(Wz_k+Ux+b)\right). $$
Peaceman-Rachford alternates a proximal reflection with the resolvent
$$ \left((1+a)I-aW\right)^{-1}. $$
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config = SolverConfig(
solver="picard", max_iter=180, tol=1e-8, backward_mode="unrolled"
)
results = {}
for splitting in ("forward_backward", "peaceman_rachford"):
model = SILVAMonotoneOperatorEquilibrium(
4,
6,
2,
operator=operator,
source=source,
prox=torch.relu,
readout=readout,
splitting=splitting,
step_size=0.5,
config=config,
)
results[splitting] = model(data.inputs, return_result=True)
state_error = torch.linalg.vector_norm(
results[splitting].state - data.equilibrium
)
print(splitting, "state error", float(state_error))
assert state_error < 2e-4
agreement = torch.linalg.vector_norm(
results["forward_backward"].state
- results["peaceman_rachford"].state
)
assert agreement < 2e-4
print("splitter agreement:", float(agreement))
config = SolverConfig(
solver="picard", max_iter=180, tol=1e-8, backward_mode="unrolled"
)
results = {}
for splitting in ("forward_backward", "peaceman_rachford"):
model = SILVAMonotoneOperatorEquilibrium(
4,
6,
2,
operator=operator,
source=source,
prox=torch.relu,
readout=readout,
splitting=splitting,
step_size=0.5,
config=config,
)
results[splitting] = model(data.inputs, return_result=True)
state_error = torch.linalg.vector_norm(
results[splitting].state - data.equilibrium
)
print(splitting, "state error", float(state_error))
assert state_error < 2e-4
agreement = torch.linalg.vector_norm(
results["forward_backward"].state
- results["peaceman_rachford"].state
)
assert agreement < 2e-4
print("splitter agreement:", float(agreement))
forward_backward state error 5.649682748298801e-07 peaceman_rachford state error 1.636992806197668e-06 splitter agreement: 1.8010382518696133e-06
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figure, axes = plt.subplots(1, 2, figsize=(7.4, 2.8))
for name, result in results.items():
axes[0].semilogy(result.solver_result.residuals, label=name.replace("_", " "))
axes[0].set(xlabel="iteration", ylabel="fixed-point residual")
axes[0].legend(fontsize=7)
target = data.target.detach().flatten()
prediction = results["peaceman_rachford"].output.detach().flatten()
axes[1].scatter(target, prediction, s=12)
limits = [float(min(target.min(), prediction.min())), float(max(target.max(), prediction.max()))]
axes[1].plot(limits, limits, color="black", linewidth=0.8)
axes[1].set(xlabel="known target", ylabel="SILVA readout")
figure.tight_layout()
plt.show()
figure, axes = plt.subplots(1, 2, figsize=(7.4, 2.8))
for name, result in results.items():
axes[0].semilogy(result.solver_result.residuals, label=name.replace("_", " "))
axes[0].set(xlabel="iteration", ylabel="fixed-point residual")
axes[0].legend(fontsize=7)
target = data.target.detach().flatten()
prediction = results["peaceman_rachford"].output.detach().flatten()
axes[1].scatter(target, prediction, s=12)
limits = [float(min(target.min(), prediction.min())), float(max(target.max(), prediction.max()))]
axes[1].plot(limits, limits, color="black", linewidth=0.8)
axes[1].set(xlabel="known target", ylabel="SILVA readout")
figure.tight_layout()
plt.show()
3. Extension Boundary¶
A new monotone architecture can replace the dense operator when it implements
forward, resolvent, and monotonicity_certificate. Convolutional,
multiscale, diagonalizable, and matrix-free resolvents can therefore use the
same source, readout, splitting, result object, and training loop.
4. Attributed CIFAR-10 Mechanism Check¶
The preceding known-solution problem answers whether the splitting is implemented correctly. This section answers a different question: can the same public constructor receive real image tensors and produce trainable logits? The ten-image snapshot contains one source-indexed CIFAR-10 example per class [81]. It is a data-path and gradient check, not a CIFAR-10 accuracy result.
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from urllib.request import urlretrieve
from silva_networks import load_source_snapshot
snapshot_path = root / "docs/assets/source-data/cifar10-balanced-10.pt"
if not snapshot_path.exists():
snapshot_path = Path(".silva-source-data") / "cifar10-balanced-10.pt"
snapshot_path.parent.mkdir(parents=True, exist_ok=True)
snapshot_url = (
"https://raw.githubusercontent.com/jseluis/silva-networks/main/"
"docs/assets/source-data/cifar10-balanced-10.pt"
)
urlretrieve(snapshot_url, snapshot_path)
source_sample = load_source_snapshot(snapshot_path)
print("dataset:", source_sample.receipt.dataset)
print("source indices:", source_sample.receipt.selected_indices)
print("content SHA-256:", source_sample.receipt.content_sha256)
print("preprocessing:")
for step in source_sample.receipt.preprocessing:
print(" -", step)
from urllib.request import urlretrieve
from silva_networks import load_source_snapshot
snapshot_path = root / "docs/assets/source-data/cifar10-balanced-10.pt"
if not snapshot_path.exists():
snapshot_path = Path(".silva-source-data") / "cifar10-balanced-10.pt"
snapshot_path.parent.mkdir(parents=True, exist_ok=True)
snapshot_url = (
"https://raw.githubusercontent.com/jseluis/silva-networks/main/"
"docs/assets/source-data/cifar10-balanced-10.pt"
)
urlretrieve(snapshot_url, snapshot_path)
source_sample = load_source_snapshot(snapshot_path)
print("dataset:", source_sample.receipt.dataset)
print("source indices:", source_sample.receipt.selected_indices)
print("content SHA-256:", source_sample.receipt.content_sha256)
print("preprocessing:")
for step in source_sample.receipt.preprocessing:
print(" -", step)
dataset: CIFAR10 source indices: (38683, 42292, 41716, 41053, 14490, 10657, 14443, 46034, 32019, 43901) content SHA-256: 2c9f290825dc8690b27ca159d4681cda3db2198d35352f888bc3fbed23f21dfe preprocessing: - deterministic class-balanced selection - scale image values to [0, 1] - bilinear resize to 16x16
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from torch.nn import functional as F
real_images = source_sample.tensors["images"]
real_labels = source_sample.tensors["labels"].long()
real_vectors = real_images.flatten(1)
real_model = SILVAMonotoneOperatorEquilibrium(
real_vectors.shape[1],
24,
10,
step_size=0.5,
splitting="forward_backward",
config=SolverConfig(
solver="picard",
max_iter=80,
tol=1e-6,
backward_mode="unrolled",
anderson_batch_dims=1,
),
)
optimizer = torch.optim.Adam(real_model.parameters(), lr=5e-3)
real_losses = []
for _ in range(3):
optimizer.zero_grad()
real_result = real_model(real_vectors, return_result=True)
real_loss = F.cross_entropy(real_result.output, real_labels)
real_loss.backward()
optimizer.step()
real_losses.append(float(real_loss.detach()))
real_result = real_model(real_vectors, return_result=True)
print("loss trajectory:", real_losses)
print("final residual:", real_result.solver_result.residual)
print("monotonicity certificate:", float(real_result.monotonicity_certificate))
assert torch.isfinite(real_result.output).all()
assert real_result.monotonicity_certificate > 0
from torch.nn import functional as F
real_images = source_sample.tensors["images"]
real_labels = source_sample.tensors["labels"].long()
real_vectors = real_images.flatten(1)
real_model = SILVAMonotoneOperatorEquilibrium(
real_vectors.shape[1],
24,
10,
step_size=0.5,
splitting="forward_backward",
config=SolverConfig(
solver="picard",
max_iter=80,
tol=1e-6,
backward_mode="unrolled",
anderson_batch_dims=1,
),
)
optimizer = torch.optim.Adam(real_model.parameters(), lr=5e-3)
real_losses = []
for _ in range(3):
optimizer.zero_grad()
real_result = real_model(real_vectors, return_result=True)
real_loss = F.cross_entropy(real_result.output, real_labels)
real_loss.backward()
optimizer.step()
real_losses.append(float(real_loss.detach()))
real_result = real_model(real_vectors, return_result=True)
print("loss trajectory:", real_losses)
print("final residual:", real_result.solver_result.residual)
print("monotonicity certificate:", float(real_result.monotonicity_certificate))
assert torch.isfinite(real_result.output).all()
assert real_result.monotonicity_certificate > 0
loss trajectory: [2.3380255699157715, 2.291635036468506, 2.19087290763855] final residual: 6.078504952711228e-07 monotonicity certificate: 1.0000743865966797
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figure, axes = plt.subplots(1, 3, figsize=(8.4, 2.5))
axes[0].imshow(real_images[0].permute(1, 2, 0))
axes[0].set_title(f"CIFAR-10 label {int(real_labels[0])}")
axes[0].axis("off")
axes[1].plot(range(1, len(real_losses) + 1), real_losses, marker="o")
axes[1].set(xlabel="optimizer step", ylabel="cross entropy")
axes[2].semilogy(real_result.solver_result.residuals)
axes[2].set(xlabel="equilibrium iteration", ylabel="residual")
figure.tight_layout()
plt.show()
figure, axes = plt.subplots(1, 3, figsize=(8.4, 2.5))
axes[0].imshow(real_images[0].permute(1, 2, 0))
axes[0].set_title(f"CIFAR-10 label {int(real_labels[0])}")
axes[0].axis("off")
axes[1].plot(range(1, len(real_losses) + 1), real_losses, marker="o")
axes[1].set(xlabel="optimizer step", ylabel="cross entropy")
axes[2].semilogy(real_result.solver_result.residuals)
axes[2].set(xlabel="equilibrium iteration", ylabel="residual")
figure.tight_layout()
plt.show()
Source-Scale Reproduction Contract¶
The compact run above verifies the defining mechanism, shapes, diagnostics, and gradients. A published benchmark requires the source data, preprocessing, architecture dimensions, optimization schedule, seeds, and evaluation budget. The executable registry keeps those obligations beside the constructor.
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from silva_networks import silva_reproduction_spec
spec = silva_reproduction_spec('silva_monotone_operator_equilibrium')
print("equation:", spec.equation)
print("datasets:", spec.datasets)
print("data sources:")
for value in spec.data_sources:
print(" -", value)
print("source-scale steps:")
for index, value in enumerate(spec.source_scale_steps, start=1):
print(f" {index}. {value}")
print("metrics:", spec.metrics)
print("preserved mechanisms:", spec.preserved_mechanisms)
print("SILVA extension points:", spec.silva_extensions)
print("benchmark obligations:", spec.benchmark_requirements)
print("constructor:", spec.constructor_signature)
from silva_networks import silva_reproduction_spec
spec = silva_reproduction_spec('silva_monotone_operator_equilibrium')
print("equation:", spec.equation)
print("datasets:", spec.datasets)
print("data sources:")
for value in spec.data_sources:
print(" -", value)
print("source-scale steps:")
for index, value in enumerate(spec.source_scale_steps, start=1):
print(f" {index}. {value}")
print("metrics:", spec.metrics)
print("preserved mechanisms:", spec.preserved_mechanisms)
print("SILVA extension points:", spec.silva_extensions)
print("benchmark obligations:", spec.benchmark_requirements)
print("constructor:", spec.constructor_signature)
equation: 0 in (I-W)z_star-Ux-b+partial f(z_star); W=(1-m)I-A^T A+B-B^T
datasets: ('MNIST', 'CIFAR-10', 'SVHN', 'compact known-solution monotone inclusions')
data sources:
- https://arxiv.org/abs/2006.08591
- https://github.com/locuslab/monotone_op_net
source-scale steps:
1. Acquire one source benchmark and reproduce its split, normalization, augmentation, and architecture dimensions.
2. Choose the forward-backward or Peaceman-Rachford route and match the monotone factorization, proximal map, step, and solver tolerances.
3. Validate the compact known-solution case, then report task accuracy, certificate, residual, evaluations, runtime, and memory at source scale.
metrics: ('task accuracy', 'monotonicity certificate', 'fixed-point residual', 'operator evaluations', 'runtime and memory')
preserved mechanisms: ('strongly monotone parameterization W=(1-m)I-A^T A+B-B^T', 'forward-backward and Peaceman-Rachford operator splittings', 'proximal nonlinearities and implicit differentiation at the solved equilibrium')
SILVA extension points: ('replace the source, proximal map, monotone operator, splitter, readout, or solver', 'inspect the monotonicity margin and numerical residual on every solve')
benchmark obligations: ('source architecture width/depth, convolutional parameterization, data split, and augmentation', 'splitting step size, forward/backward tolerances, optimizer, regularization, and seeds', 'task accuracy, residual, evaluation count, memory, and source baselines')
constructor: (in_dim: 'int', state_dim: 'int', out_dim: 'int', *, operator: 'nn.Module | None' = None, source: 'nn.Module | None' = None, prox: 'Callable[[Tensor], Tensor]' = <function relu>, readout: 'nn.Module | None' = None, splitting: 'MonotoneSplitting' = 'forward_backward', step_size: 'float' = 1.0, margin: 'float' = 1.0, config: 'SolverConfig | None' = None) -> 'None'
From 36 Silva Monotone Operator 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.
Replace the Monotone Operator, Proximal Map, and Resolvent¶
model = SILVAMonotoneOperatorEquilibrium(
in_dim=input_dim,
state_dim=state_dim,
out_dim=output_dim,
operator=my_monotone_operator,
source=my_source,
prox=my_step_aware_prox,
readout=my_readout,
splitting="peaceman_rachford",
step_size=step_size,
config=solver_config,
)
The custom operator supplies forward, resolvent, and
monotonicity_certificate. Dense, convolutional, and multiscale versions use
the same three-method contract; source-scale convolutional runs must also
preserve the Fourier-domain resolvent and the paper's backward splitting.
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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 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.
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notebook_reproduction_record = {
"notebook": '36_silva_monotone_operator_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": '36_silva_monotone_operator_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[11]:
{'notebook': '36_silva_monotone_operator_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 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.
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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('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 [13]:
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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('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.