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SILVA Physics Graph Equilibrium: Transport Dataset Lab¶

This lab derives graph diffusion and directed transport, verifies a generated steady-state dataset, trains a node-level SILVA equilibrium, and tests node relabeling. The canonical family is silva_physics_graph_deq [44].

Numbered literature: [1], [36], [44]. Each number opens the complete citation and its primary external source.

In [1]:

from pathlib import Path
import importlib.util
import subprocess
import sys

IN_HOSTED_RUNTIME = "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")]
    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_HOSTED_RUNTIME 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_HOSTED_RUNTIME = "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")] 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_HOSTED_RUNTIME and importlib.util.find_spec("silva_networks") is None: subprocess.check_call([sys.executable, "-m", "pip", "install", f"git+{REPO_URL}"]) root = Path.cwd() else: root = Path.cwd()

In [2]:

import torch
import matplotlib.pyplot as plt

from silva_networks import (
    SILVAPhysicsGuidedGraphDEQ,
    SolverConfig,
    graph_convection_diffusion,
    make_graph_transport_dataset,
    silva_equilibrium_model,
)

plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
torch.manual_seed(180)

import torch import matplotlib.pyplot as plt from silva_networks import ( SILVAPhysicsGuidedGraphDEQ, SolverConfig, graph_convection_diffusion, make_graph_transport_dataset, silva_equilibrium_model, ) plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300}) torch.manual_seed(180)

Out[2]:

<torch._C.Generator at 0x10ce5ceb0>

1. Continuous and Discrete Transport¶

A steady convection-diffusion-reaction equation can be written

$$ 0=s+\gamma_r u+\gamma_d\Delta u -\gamma_a\mathbf v\cdot\nabla u-u. $$

For incoming edge $i\rightarrow j$, SILVA uses

$$ (\mathcal L_GZ)_j =\frac1{d_j}\sum_{i\rightarrow j} w_{ij}(Z_i-Z_j), $$

$$ (\nabla_VZ)_j =\frac1{d_j}\sum_{i\rightarrow j} v_{ij}(Z_j-Z_i). $$

The first is symmetric when conductances and reverse edges agree. The second retains orientation through the signed edge velocity [44].

In [3]:

data = make_graph_transport_dataset(
    samples=4,
    nodes=10,
    reaction_scale=0.05,
    diffusion_scale=0.2,
    advection_scale=0.05,
    seed=18,
)
physical_error = data.equation_residual().abs().max()

assert data.x.shape == (40, 3)
assert data.edge_index.shape == (2, 80)
assert physical_error < 2e-6
print("nodes:", data.x.shape[0])
print("graphs:", int(data.batch.max()) + 1)
print("maximum dataset equation residual:", float(physical_error))

data = make_graph_transport_dataset( samples=4, nodes=10, reaction_scale=0.05, diffusion_scale=0.2, advection_scale=0.05, seed=18, ) physical_error = data.equation_residual().abs().max() assert data.x.shape == (40, 3) assert data.edge_index.shape == (2, 80) assert physical_error < 2e-6 print("nodes:", data.x.shape[0]) print("graphs:", int(data.batch.max()) + 1) print("maximum dataset equation residual:", float(physical_error))

nodes: 40
graphs: 4
maximum dataset equation residual: 1.1920928955078125e-07

2. SILVA Branch Equation¶

Node observations $X$ enter the source branch. Reaction, diffusion, and advection receive distinct learned channel maps:

$$ \begin{aligned} T(Z;X) &=\phi\left[S(X)+\gamma_rR(Z)\right.\\ &\qquad+\gamma_dD(\mathcal L_GZ)\\ &\qquad\left.-\gamma_aA(\nabla_VZ)\right]. \end{aligned} $$

The node equilibrium is $Z^\star=T(Z^\star;X)$. A node readout predicts one value per node; graph pooling is applied only when the task has one target per graph.

In [4]:

probe = torch.arange(10, dtype=torch.float32).unsqueeze(-1)
one_graph_edges = data.edge_index[:, :20]
diffusion, gradient = graph_convection_diffusion(
    probe,
    one_graph_edges,
    edge_weight=data.edge_weight[:20],
    edge_velocity=data.edge_velocity[:20],
)
print("diffusion shape:", tuple(diffusion.shape))
print("directed-gradient shape:", tuple(gradient.shape))
print("constant-field diffusion check:", float(
    graph_convection_diffusion(torch.ones_like(probe), one_graph_edges)[0].abs().max()
))

probe = torch.arange(10, dtype=torch.float32).unsqueeze(-1) one_graph_edges = data.edge_index[:, :20] diffusion, gradient = graph_convection_diffusion( probe, one_graph_edges, edge_weight=data.edge_weight[:20], edge_velocity=data.edge_velocity[:20], ) print("diffusion shape:", tuple(diffusion.shape)) print("directed-gradient shape:", tuple(gradient.shape)) print("constant-field diffusion check:", float( graph_convection_diffusion(torch.ones_like(probe), one_graph_edges)[0].abs().max() ))

diffusion shape: (10, 1)
directed-gradient shape: (10, 1)
constant-field diffusion check: 0.0

3. Train a Small Node Field¶

The dataset target solves the discrete linear equation exactly. The learned transition is nonlinear, so training asks the SILVA equilibrium and readout to approximate that solution operator over four source fields.

In [5]:

model = SILVAPhysicsGuidedGraphDEQ(
    in_dim=3,
    state_dim=6,
    out_dim=1,
    config=SolverConfig(
        solver="picard",
        max_iter=7,
        tol=1e-5,
        alpha=0.7,
        backward_mode="unrolled",
    ),
)
optimizer = torch.optim.Adam(model.parameters(), lr=4e-3)
losses = []
for epoch in range(6):
    optimizer.zero_grad()
    prediction = model(
        data.x,
        data.edge_index,
        edge_weight=data.edge_weight,
        edge_velocity=data.edge_velocity,
    )
    loss = torch.nn.functional.mse_loss(prediction, data.target)
    loss.backward()
    optimizer.step()
    losses.append(float(loss.detach()))

result = model(
    data.x,
    data.edge_index,
    edge_weight=data.edge_weight,
    edge_velocity=data.edge_velocity,
    return_result=True,
)
assert result.output.shape == data.target.shape
assert all(torch.isfinite(torch.tensor(losses)))
print("training losses:", losses)
print("fixed-point residual:", result.solver_result.residual)
print("prediction physical residual:", float(
    data.equation_residual(result.output).square().mean().sqrt()
))

model = SILVAPhysicsGuidedGraphDEQ( in_dim=3, state_dim=6, out_dim=1, config=SolverConfig( solver="picard", max_iter=7, tol=1e-5, alpha=0.7, backward_mode="unrolled", ), ) optimizer = torch.optim.Adam(model.parameters(), lr=4e-3) losses = [] for epoch in range(6): optimizer.zero_grad() prediction = model( data.x, data.edge_index, edge_weight=data.edge_weight, edge_velocity=data.edge_velocity, ) loss = torch.nn.functional.mse_loss(prediction, data.target) loss.backward() optimizer.step() losses.append(float(loss.detach())) result = model( data.x, data.edge_index, edge_weight=data.edge_weight, edge_velocity=data.edge_velocity, return_result=True, ) assert result.output.shape == data.target.shape assert all(torch.isfinite(torch.tensor(losses))) print("training losses:", losses) print("fixed-point residual:", result.solver_result.residual) print("prediction physical residual:", float( data.equation_residual(result.output).square().mean().sqrt() ))

training losses: [0.7732363939285278, 0.7518482804298401, 0.7311099767684937, 0.7110251188278198, 0.691594123840332, 0.6728144884109497]
fixed-point residual: 0.00538542540743947
prediction physical residual: 0.8046050667762756

In [6]:

nodes = 10
fig, axes = plt.subplots(1, 3, figsize=(8.2, 2.4))
axes[0].plot(data.coordinates[:nodes, 0], data.x[:nodes, 0], marker="o")
axes[0].set_title("source")
axes[1].plot(data.coordinates[:nodes, 0], data.target[:nodes, 0], label="exact")
axes[1].plot(
    data.coordinates[:nodes, 0],
    result.output[:nodes, 0].detach(),
    "--",
    label="SILVA",
)
axes[1].legend()
axes[1].set_title("steady field")
axes[2].plot(range(1, len(losses) + 1), losses, marker="o")
axes[2].set_yscale("log")
axes[2].set_xlabel("epoch")
axes[2].set_title("training loss")
fig.tight_layout()
plt.show()

nodes = 10 fig, axes = plt.subplots(1, 3, figsize=(8.2, 2.4)) axes[0].plot(data.coordinates[:nodes, 0], data.x[:nodes, 0], marker="o") axes[0].set_title("source") axes[1].plot(data.coordinates[:nodes, 0], data.target[:nodes, 0], label="exact") axes[1].plot( data.coordinates[:nodes, 0], result.output[:nodes, 0].detach(), "--", label="SILVA", ) axes[1].legend() axes[1].set_title("steady field") axes[2].plot(range(1, len(losses) + 1), losses, marker="o") axes[2].set_yscale("log") axes[2].set_xlabel("epoch") axes[2].set_title("training loss") fig.tight_layout() plt.show()

4. Node Relabeling¶

Let $P$ be a node permutation. A graph transition should satisfy

$$ T(PZ;PX,PE)=P\,T(Z;X,E), $$

where $PE$ means that both rows of edge_index are relabeled consistently. This is a structural test, not a statistical expectation.

In [7]:

single_x = data.x[:10]
single_edges = data.edge_index[:, :20]
single_weight = data.edge_weight[:20]
single_velocity = data.edge_velocity[:20]
baseline = model(
    single_x,
    single_edges,
    edge_weight=single_weight,
    edge_velocity=single_velocity,
)
permutation = torch.tensor([3, 0, 8, 1, 6, 2, 9, 5, 7, 4])
inverse = torch.empty_like(permutation)
inverse[permutation] = torch.arange(permutation.numel())
permuted_edges = inverse[single_edges]
permuted = model(
    single_x[permutation],
    permuted_edges,
    edge_weight=single_weight,
    edge_velocity=single_velocity,
)
relabel_error = (permuted - baseline[permutation]).abs().max()
assert relabel_error < 2e-5
print("node relabeling error:", float(relabel_error))

single_x = data.x[:10] single_edges = data.edge_index[:, :20] single_weight = data.edge_weight[:20] single_velocity = data.edge_velocity[:20] baseline = model( single_x, single_edges, edge_weight=single_weight, edge_velocity=single_velocity, ) permutation = torch.tensor([3, 0, 8, 1, 6, 2, 9, 5, 7, 4]) inverse = torch.empty_like(permutation) inverse[permutation] = torch.arange(permutation.numel()) permuted_edges = inverse[single_edges] permuted = model( single_x[permutation], permuted_edges, edge_weight=single_weight, edge_velocity=single_velocity, ) relabel_error = (permuted - baseline[permutation]).abs().max() assert relabel_error < 2e-5 print("node relabeling error:", float(relabel_error))

node relabeling error: 0.0

5. Node Tasks and Graph Tasks¶

For node prediction, the readout receives every equilibrium node. For graph prediction, SILVA first computes a mask-free mean, sum, or max pooling inside each graph id. Choose pooling from the units of the target: means are intensive, sums are extensive, and maxima represent extremes.

In [8]:

graph_model = silva_equilibrium_model(
    "silva_physics_graph_deq",
    in_dim=3,
    state_dim=4,
    out_dim=1,
    task="graph",
    pooling="mean",
    config=SolverConfig(max_iter=4, alpha=0.7),
)
graph_values = graph_model(
    data.x,
    data.edge_index,
    edge_weight=data.edge_weight,
    edge_velocity=data.edge_velocity,
    batch=data.batch,
)
assert graph_values.shape == (4, 1)
print("graph-level output:", tuple(graph_values.shape))

graph_model = silva_equilibrium_model( "silva_physics_graph_deq", in_dim=3, state_dim=4, out_dim=1, task="graph", pooling="mean", config=SolverConfig(max_iter=4, alpha=0.7), ) graph_values = graph_model( data.x, data.edge_index, edge_weight=data.edge_weight, edge_velocity=data.edge_velocity, batch=data.batch, ) assert graph_values.shape == (4, 1) print("graph-level output:", tuple(graph_values.shape))

graph-level output: (4, 1)

6. Practical Guidance¶

Problem	Diagnostic	Response
transport direction is reversed	inspect one directed edge by hand	document source -> destination and velocity sign
node values oversmooth	compare reaction, diffusion, and transport ablations	reduce diffusion or retain source injection
graph residual is low but physics residual is high	evaluate the discrete equation separately	constrain or supervise the physical branches
batched graphs interact accidentally	compare graph ids at both edge endpoints	offset edges and validate every batch

The small ring dataset tests the implementation and physical bookkeeping. Environmental benchmark claims require the measurement network, missing-data rules, temporal split, and preprocessing used by the cited study [44].

From 18 Silva Graph Transport Lab 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	one latent vector per node or entity
Condition	node features, edges, edge attributes, and graph batches
Repeated computation	a source-injected graph message or monotone graph transition
Required invariants	node relabeling equivariance, graph boundaries, and state shape
Replaceable components	input projection, message field, global field, transition, pooling, and head

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 Graph Physics Transition¶

class MyGraphPhysics(nn.Module):
    def forward(
        self, state, inputs, edge_index, *, edge_weight=None, edge_velocity=None
    ):
        source = self.source(inputs)
        transport = self.transport(state, edge_index, edge_velocity)
        diffusion = self.diffusion(state, edge_index, edge_weight)
        return torch.tanh(source + transport + diffusion)


model = SILVAPhysicsGuidedGraphDEQ(
    in_dim=input_dim,
    state_dim=width,
    out_dim=output_dim,
    transition=MyGraphPhysics(),
    readout=my_readout,
    config=solver_config,
)

In [9]:

silva-extension-curriculum

import torch as silva_extension_torch
from torch import nn as silva_extension_nn

from silva_networks import (
    SILVAConditionedEquilibrium,
    SILVAZeroInitializer,
    SolverConfig,
    validate_silva_transition,
)


class NotebookExtensionTransition(silva_extension_nn.Module):
    def __init__(self, condition_dim=2, state_dim=3):
        super().__init__()
        self.source = silva_extension_nn.Linear(condition_dim, state_dim)
        self.state_field = silva_extension_nn.Sequential(
            silva_extension_nn.Linear(state_dim, 2 * state_dim),
            silva_extension_nn.Tanh(),
            silva_extension_nn.Linear(2 * state_dim, state_dim),
        )

    def forward(self, state, condition):
        return silva_extension_torch.tanh(
            self.source(condition) + 0.15 * self.state_field(state)
        )


silva_extension_torch.manual_seed(610)
notebook_condition = silva_extension_torch.linspace(-1.0, 1.0, 8).reshape(4, 2)
notebook_state0 = silva_extension_torch.zeros(4, 3)
notebook_transition = NotebookExtensionTransition()

notebook_report = validate_silva_transition(
    notebook_transition,
    notebook_state0,
    notebook_condition,
)
assert notebook_report.valid

with silva_extension_torch.no_grad():
    notebook_reference_step = silva_extension_torch.tanh(
        notebook_transition.source(notebook_condition)
        + 0.15 * notebook_transition.state_field(notebook_state0)
    )
silva_extension_torch.testing.assert_close(
    notebook_transition(notebook_state0, notebook_condition),
    notebook_reference_step,
)

notebook_custom_model = SILVAConditionedEquilibrium(
    notebook_transition,
    SILVAZeroInitializer(3),
    readout=silva_extension_nn.Linear(3, 1),
    config=SolverConfig(
        solver="picard",
        max_iter=40,
        tol=1e-7,
        backward_mode="implicit",
        backward_solver="gmres",
        anderson_batch_dims=1,
    ),
)
notebook_custom_result = notebook_custom_model(
    notebook_condition,
    return_result=True,
)
assert notebook_custom_result.output.shape == (4, 1)
assert notebook_custom_result.solver_result.residual < 1e-5

notebook_custom_result.output.square().mean().backward()
assert all(
    parameter.grad is not None and silva_extension_torch.isfinite(parameter.grad).all()
    for parameter in notebook_custom_model.parameters()
)
print("custom transition:", notebook_report)
print("equilibrium residual:", notebook_custom_result.solver_result.residual)

import torch as silva_extension_torch from torch import nn as silva_extension_nn from silva_networks import ( SILVAConditionedEquilibrium, SILVAZeroInitializer, SolverConfig, validate_silva_transition, ) class NotebookExtensionTransition(silva_extension_nn.Module): def __init__(self, condition_dim=2, state_dim=3): super().__init__() self.source = silva_extension_nn.Linear(condition_dim, state_dim) self.state_field = silva_extension_nn.Sequential( silva_extension_nn.Linear(state_dim, 2 * state_dim), silva_extension_nn.Tanh(), silva_extension_nn.Linear(2 * state_dim, state_dim), ) def forward(self, state, condition): return silva_extension_torch.tanh( self.source(condition) + 0.15 * self.state_field(state) ) silva_extension_torch.manual_seed(610) notebook_condition = silva_extension_torch.linspace(-1.0, 1.0, 8).reshape(4, 2) notebook_state0 = silva_extension_torch.zeros(4, 3) notebook_transition = NotebookExtensionTransition() notebook_report = validate_silva_transition( notebook_transition, notebook_state0, notebook_condition, ) assert notebook_report.valid with silva_extension_torch.no_grad(): notebook_reference_step = silva_extension_torch.tanh( notebook_transition.source(notebook_condition) + 0.15 * notebook_transition.state_field(notebook_state0) ) silva_extension_torch.testing.assert_close( notebook_transition(notebook_state0, notebook_condition), notebook_reference_step, ) notebook_custom_model = SILVAConditionedEquilibrium( notebook_transition, SILVAZeroInitializer(3), readout=silva_extension_nn.Linear(3, 1), config=SolverConfig( solver="picard", max_iter=40, tol=1e-7, backward_mode="implicit", backward_solver="gmres", anderson_batch_dims=1, ), ) notebook_custom_result = notebook_custom_model( notebook_condition, return_result=True, ) assert notebook_custom_result.output.shape == (4, 1) assert notebook_custom_result.solver_result.residual < 1e-5 notebook_custom_result.output.square().mean().backward() assert all( parameter.grad is not None and silva_extension_torch.isfinite(parameter.grad).all() for parameter in notebook_custom_model.parameters() ) print("custom transition:", notebook_report) print("equilibrium residual:", notebook_custom_result.solver_result.residual)

custom transition: SILVATransitionReport(state_shape=(4, 3), output_shape=(4, 3), preserves_shape=True, preserves_device=True, preserves_dtype=True, finite=True, differentiable=True, parameter_count=54)
equilibrium residual: 5.960464477539063e-08

Numerical Equivalence, Compact Reproduction, and Scale¶

Before training, compare one packaged transition with an independently written update:

$$ e_{\mathrm{step}} =\frac{\|T_\theta(z,x)-T_{\mathrm{ref}}(z,x)\|_2} {\|T_{\mathrm{ref}}(z,x)\|_2+\varepsilon}. $$

After solving, report the fixed-point residual separately:

$$ e_{\mathrm{fp}} =\frac{\|T_\theta(z^\star,x)-z^\star\|_2} {\|z^\star\|_2+\varepsilon}. $$

For this notebook, a compact reproduction must declare and assert node/graph error, physical graph residual, 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 node count, edge count, feature width, and number of graphs. Increase one axis at a time, retain the compact deterministic case as a regression test, and record task error, domain-specific residual, forward residual, backward linear residual, memory use, and runtime independently.

Extension Exercises¶

1. Replace one component from this notebook while preserving its state and domain invariants.
2. Write the replacement first as an independent reference function, then as a module, and assert one-step equivalence.
3. Compare two solver configurations on the identical trained transition.
4. Add a compact baseline and a predeclared metric threshold.
5. Create a full-scale configuration without weakening the compact tests.

The complete authoring protocol is documented in Extending SILVA.

In [10]:

silva-extension-curriculum

notebook_reproduction_record = {
    "notebook": '18_silva_graph_transport_lab.ipynb',
    "state": 'one latent vector per node or entity',
    "condition": 'node features, edges, edge attributes, and graph batches',
    "transition": 'a source-injected graph message or monotone graph transition',
    "invariants": 'node relabeling equivariance, graph boundaries, and state shape',
    "compact_metric": 'node/graph error, physical graph residual, and fixed-point residual',
    "scale_axis": 'node count, edge count, feature width, and number of graphs',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record

notebook_reproduction_record = { "notebook": '18_silva_graph_transport_lab.ipynb', "state": 'one latent vector per node or entity', "condition": 'node features, edges, edge attributes, and graph batches', "transition": 'a source-injected graph message or monotone graph transition', "invariants": 'node relabeling equivariance, graph boundaries, and state shape', "compact_metric": 'node/graph error, physical graph residual, and fixed-point residual', "scale_axis": 'node count, edge count, feature width, and number of graphs', } assert all(notebook_reproduction_record.values()) notebook_reproduction_record

Out[10]:

{'notebook': '18_silva_graph_transport_lab.ipynb',
 'state': 'one latent vector per node or entity',
 'condition': 'node features, edges, edge attributes, and graph batches',
 'transition': 'a source-injected graph message or monotone graph transition',
 'invariants': 'node relabeling equivariance, graph boundaries, and state shape',
 'compact_metric': 'node/graph error, physical graph residual, and fixed-point residual',
 'scale_axis': 'node count, edge count, feature width, and number of graphs'}

Worked Convergence and Sensitivity Study¶

The preceding example demonstrates one configured solve. This additional study changes the graph propagation 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, one latent vector per node or entity, and its repeated map, a source-injected graph message or monotone graph transition. The scalar study does not replace the domain model; it supplies a result whose convergence rate and derivative are known exactly, so the same reporting code can be trusted before it is applied to the larger transition.

In [11]:

silva-extension-curriculum

import math as silva_deepening_math
import torch as silva_deepening_torch

silva_deepening_rates = (0.20, 0.45, 0.70, 0.85)
silva_deepening_source = 0.35
silva_deepening_tolerance = 1e-8
silva_deepening_histories = {}
silva_deepening_rows = []

for silva_deepening_rho in silva_deepening_rates:
    silva_deepening_state = silva_deepening_torch.tensor(0.0)
    silva_deepening_exact = silva_deepening_source / (1.0 - silva_deepening_rho)
    silva_deepening_history = []
    for silva_deepening_iteration in range(1, 241):
        silva_deepening_next = (
            silva_deepening_rho * silva_deepening_state + silva_deepening_source
        )
        silva_deepening_residual = abs(
            float(silva_deepening_next - silva_deepening_state)
        )
        silva_deepening_history.append(silva_deepening_residual)
        silva_deepening_state = silva_deepening_next
        if silva_deepening_residual < silva_deepening_tolerance:
            break

    silva_deepening_u = silva_deepening_torch.tensor(
        silva_deepening_source, requires_grad=True
    )
    silva_deepening_solution = silva_deepening_u / (1.0 - silva_deepening_rho)
    silva_deepening_solution.backward()
    silva_deepening_expected_sensitivity = 1.0 / (1.0 - silva_deepening_rho)
    silva_deepening_gradient_error = abs(
        float(silva_deepening_u.grad) - silva_deepening_expected_sensitivity
    )
    silva_deepening_histories[silva_deepening_rho] = silva_deepening_history
    silva_deepening_rows.append(
        (
            silva_deepening_rho,
            silva_deepening_iteration,
            silva_deepening_history[-1],
            abs(float(silva_deepening_state) - silva_deepening_exact),
            float(silva_deepening_u.grad),
            silva_deepening_gradient_error,
        )
    )

print('graph propagation 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('graph propagation 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)

graph propagation factor
rho | iterations | final residual | exact-state error | sensitivity | gradient error
0.20 |  12 | 0.000e+00 | 5.551e-17 | 1.2500 | 0.000e+00
0.45 |  23 | 0.000e+00 | 1.084e-08 | 1.8182 | 6.502e-08
0.70 |  45 | 0.000e+00 | 1.589e-07 | 3.3333 | 7.947e-08
0.85 |  93 | 0.000e+00 | 5.563e-07 | 6.6667 | 1.589e-07

In [12]:

silva-extension-curriculum

import matplotlib.pyplot as silva_deepening_plt

silva_deepening_plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
silva_deepening_figure, silva_deepening_axes = silva_deepening_plt.subplots(
    1, 2, figsize=(8.6, 3.2)
)
for silva_deepening_rho, silva_deepening_history in silva_deepening_histories.items():
    silva_deepening_axes[0].semilogy(
        range(1, len(silva_deepening_history) + 1),
        silva_deepening_history,
        marker="o",
        markersize=2,
        linewidth=1.2,
        label=f"rho={silva_deepening_rho:.2f}",
    )
silva_deepening_axes[0].axhline(
    silva_deepening_tolerance, color="black", linestyle="--", linewidth=0.9
)
silva_deepening_axes[0].set_xlabel("iteration")
silva_deepening_axes[0].set_ylabel("absolute residual")
silva_deepening_axes[0].set_title("Residual trajectories")
silva_deepening_axes[0].legend(fontsize=7)

silva_deepening_axes[1].plot(
    [row[0] for row in silva_deepening_rows],
    [row[1] for row in silva_deepening_rows],
    marker="o",
    label="iterations",
)
silva_deepening_sensitivity_axis = silva_deepening_axes[1].twinx()
silva_deepening_sensitivity_axis.plot(
    [row[0] for row in silva_deepening_rows],
    [row[4] for row in silva_deepening_rows],
    color="tab:red",
    marker="s",
    label="sensitivity",
)
silva_deepening_axes[1].set_xlabel('graph propagation 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('graph propagation 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 graph propagation 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	node/graph error, physical graph residual, 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	node relabeling equivariance, graph boundaries, and state shape
Scale sweep	Change one of node count, edge count, feature width, and number of graphs at a time

A richer experiment should now repeat the sweep with at least two forward solvers, two tolerances, and multiple seeds. Keep model parameters and data identical when comparing solvers. Then change one architecture or data-scale axis, retain the compact analytic study as a regression test, and report task quality, residuals, iterations, runtime, memory, gradient norms, and failed convergence cases together.

Where to Go Next¶

Question	Page
How was the graph transport target constructed?	Dataset-Backed Equilibrium Labs
How are graph branches derived inside SILVA?	Recent Equilibrium Families
Which graph dataset tensors are public?	Recent Dataset API