Coupled RAFT and DEQ-Flow in SILVA¶
The equilibrium state is $(h,u)$:
$$ h^+=\operatorname{ConvGRU}(h,c,m(u,C(u))),\qquad u^+=u+\Delta_\theta(h^+). $$
This package-native case exposes residual-encoder stages and stride, correlation pyramid levels and radius, motion/GRU widths, global aggregation, solver and gradient rules, sparse correction indices, learned convex upsampling, fixed-point reuse, and custom encoder or update modules.
Numbered literature: [22], [23], [24]. Each number opens the complete citation and its primary external source.
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()
import torch
from silva_networks import (
SILVARAFTDEQ,
SolverConfig,
make_silva_translation_flow_batch,
silva_flow_fixed_point_correction_loss,
)
torch.manual_seed(13)
batch = make_silva_translation_flow_batch(
batch_size=1, channels=1, height=8, width=8, shift=(1.0, 0.0)
)
config = SolverConfig(
solver="picard",
max_iter=3,
alpha=0.5,
indexing=(1, 2),
backward_mode="implicit",
backward_solver="gmres",
backward_max_iter=8,
)
model = SILVARAFTDEQ(
in_channels=1,
feature_dim=8,
hidden_dim=4,
context_dim=4,
encoder_channels=(4,),
encoder_residual_blocks=1,
encoder_dropout=0.0,
output_stride=2,
corr_levels=2,
corr_radius=1,
motion_dim=8,
flow_head_dim=8,
gru_kernel_size=3,
correlation_hidden_dims=(8, 8),
flow_hidden_dims=(8, 4),
correction_steps=1,
config=config,
)
Solve, Sparse Corrections, and Exact Gradient¶
indexing stores selected numerical states. Short differentiable
corrections turn them into auxiliary flow predictions without
retaining the entire forward solver graph.
result = model(batch.image1, batch.image2, return_result=True)
predictions = result.flow_sequence or [result.flow]
loss = silva_flow_fixed_point_correction_loss(
predictions, batch.flow, valid=batch.valid, gamma=0.8
)
loss.backward()
finite_gradients = all(
parameter.grad is None or torch.isfinite(parameter.grad).all()
for parameter in model.parameters()
)
print("flow", result.flow.shape)
print("low-resolution state", result.low_resolution_flow.shape)
print("correction predictions", len(predictions))
print("finite gradients", bool(finite_gradients))
flow torch.Size([1, 2, 8, 8]) low-resolution state torch.Size([1, 2, 4, 4]) correction predictions 3 finite gradients True
Reuse the Fixed Point¶
A previous hidden/flow equilibrium can initialize a related image pair. Whether this is appropriate across frames or augmentations is an experiment choice.
reused = model(batch.image1, batch.image2, cached_state=result.cached_state)
print("reused flow", reused.shape)
reused flow torch.Size([1, 2, 8, 8])
Reproduction Boundary and Citations¶
This compact run validates the coupled state, correlation/GRU update, learned upsampling, correction loss, implicit backward path, and reuse contract. Paper metrics require the source dataset mixtures, augmentations, schedules, evaluation code, resolution, and model dimensions.
Cite RAFT for all-pairs correlation and recurrent refinement, DEQ-Flow for the equilibrium optical-flow formulation and sparse correction/reuse strategy, and SILVA Networks for this generalized package API: https://github.com/jseluis/silva-networks https://doi.org/10.5281/zenodo.21770098
From 13 Raft Deq Flow 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 flow field, optionally coupled to a recurrent hidden state |
| Condition | image features, correlation volumes, context, and initial flow |
| Repeated computation | the tied correlation-conditioned refinement update |
| Required invariants | flow shape, coordinate convention, image resolution, and warping domain |
| Replaceable components | feature/context encoders, correlation, update block, transition, upsampler, 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 Flow Transition¶
The complete refinement can be supplied as
transition_module(flow, fmap1, fmap2, correlation). Feature and context
encoders and the update block are independently replaceable.
flow_model = SILVAOpticalFlowDEQ(
feature_dim=feature_dim,
encoder_module=my_feature_encoder,
update_block=my_update_block,
transition_module=my_flow_transition,
config=solver_config,
)
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 endpoint error, warp error, correction loss, and fixed-point residual. 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 image resolution, correlation radius/levels, hidden width, and solver budget. 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.
notebook_reproduction_record = {
"notebook": '13_raft_deq_flow.ipynb',
"state": 'the flow field, optionally coupled to a recurrent hidden state',
"condition": 'image features, correlation volumes, context, and initial flow',
"transition": 'the tied correlation-conditioned refinement update',
"invariants": 'flow shape, coordinate convention, image resolution, and warping domain',
"compact_metric": 'endpoint error, warp error, correction loss, and fixed-point residual',
"scale_axis": 'image resolution, correlation radius/levels, hidden width, and solver budget',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record
{'notebook': '13_raft_deq_flow.ipynb',
'state': 'the flow field, optionally coupled to a recurrent hidden state',
'condition': 'image features, correlation volumes, context, and initial flow',
'transition': 'the tied correlation-conditioned refinement update',
'invariants': 'flow shape, coordinate convention, image resolution, and warping domain',
'compact_metric': 'endpoint error, warp error, correction loss, and fixed-point residual',
'scale_axis': 'image resolution, correlation radius/levels, hidden width, and solver budget'}
Worked Convergence and Sensitivity Study¶
The preceding example demonstrates one configured solve. This additional study changes the refinement 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 flow field, optionally coupled to a recurrent hidden state, and its repeated map, the tied correlation-conditioned refinement update. 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.
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('refinement 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)
refinement 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
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('refinement 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 refinement 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 | endpoint error, warp error, correction loss, and fixed-point residual |
| 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 | flow shape, coordinate convention, image resolution, and warping domain |
| Scale sweep | Change one of image resolution, correlation radius/levels, hidden width, and solver budget 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.