Dataset-Backed Equilibrium Labs

The recent SILVA families now have deterministic datasets matched to their state geometry. These builders make the governing equation or statistical contract observable before a learned model is introduced.

Family	Dataset builder	State geometry	Exact check
Fourier equilibrium	make_periodic_elliptic_dataset	regular fields	periodic elliptic residual
physics graph equilibrium	make_graph_transport_dataset	batched ring graphs	discrete steady transport residual
homotopy equilibrium	make_affine_homotopy_dataset	condition/root pairs	affine fixed-point residual
distributional equilibrium	make_variable_measure_dataset	padded empirical measures	masks, counts, and empirical moments

Each lab pairs a derivation with an executable notebook and retained 300 DPI plots.

Periodic Elliptic Fields

Open Fourier equilibrium lab Download notebook

The field dataset solves

\[ (-\Delta+m)u=f \]

on a periodic unit square. For wave vector \(k\),

\[ \widehat u(k) =\frac{\widehat f(k)}{|k|^2+m}. \]

The forcing is restricted to low modes, and the target is evaluated from this spectral formula. The returned batch contains

forcing:      (samples, 1, height, width)
target:       (samples, 1, height, width)
coordinates:  (height, width, 2)

from silva_networks import make_periodic_elliptic_dataset

data = make_periodic_elliptic_dataset(
    samples=16,
    height=32,
    width=32,
    modes=4,
    mass=1.0,
    seed=17,
)
assert data.equation_residual().abs().max() < 1e-4

This builder tests field shapes, Fourier normalization, resolution changes, and equation-aware evaluation. The full FNO-DEQ study evaluates Darcy flow and steady Navier-Stokes with its published datasets and protocols [43].

Graph Transport Fields

Open graph transport lab Download notebook

For every generated graph, the target satisfies

\[ u=s+gamma_r u +gamma_d\mathcal L_Gu -\gamma_a\nabla_Vu. \]

The ring graph has forward and reverse edges. Conductance is stored in edge_weight; direction and speed are stored in edge_velocity. Multiple graphs are packed by offsetting node ids and assigning a graph id in batch.

x:              (samples * nodes, 3)
edge_index:     (2, samples * 2 * nodes)
edge_weight:    (samples * 2 * nodes,)
edge_velocity:  (samples * 2 * nodes,)
batch:          (samples * nodes,)
target:         (samples * nodes, 1)

from silva_networks import make_graph_transport_dataset

data = make_graph_transport_dataset(samples=8, nodes=24, seed=18)
assert data.equation_residual().abs().max() < 1e-4
assert (data.batch[data.edge_index[0]] == data.batch[data.edge_index[1]]).all()

The cited environmental study uses real NO2 and PM2.5 measurements collected in Antwerp, discretized over spatial locations and hourly intervals [44]. The package ring data is for equation, batching, and training validation; environmental reporting must retain the measurement geometry, missing-data rules, split, and physical units from the study.

Affine Homotopy Pairs

Open homotopy equilibrium lab Download notebook

For transition

\[ T(z;x)=az+x, \qquad |a|<1, \]

the exact root is

\[ z^\star=\frac{x}{1-a}. \]

The residual flow also has the complete analytic trajectory

\[ z(t)=z^\star+(z_0-z^\star)e^{-(1-a)t}. \]

The builder returns conditions and exact roots. It supports endpoint error, trajectory error, Euler/RK4 comparison, gradient checks, and horizon studies.

from silva_networks import make_affine_homotopy_dataset

data = make_affine_homotopy_dataset(
    samples=64,
    dimension=4,
    contraction=0.5,
    seed=19,
)
assert data.fixed_point_residual().abs().max() < 1e-6

The corresponding research family connects equilibrium models and continuous paths through homotopy continuation [46]. The exact affine data is a numerical reference, not a replacement for the vision datasets and training protocol used in that study.

Variable-Size Empirical Measures

Open distributional equilibrium lab Download notebook

Each sample contains a variable number of points from a Gaussian mixture. The batch pads them to one maximum length and supplies a boolean mask:

context:            (samples, max_particles, dimension)
context_mask:       (samples, max_particles)
component_centers:  (samples, components, dimension)
target_mean:        (samples, dimension)
counts:             (samples,)

The mask-aware empirical mean is

\[ \overline x_b =\frac{ \sum_i m_{bi}x_{bi} }{ \sum_i m_{bi} }. \]

import torch
from silva_networks import make_variable_measure_dataset

data = make_variable_measure_dataset(
    samples=32,
    min_particles=16,
    max_particles=40,
    dimension=3,
    components=3,
    seed=20,
)
assert (data.context_mask.sum(dim=1) == data.counts).all()
assert torch.allclose(data.empirical_mean(), data.target_mean)

The distributional study evaluates point-cloud classification and completion; its maintained research materials identify MNIST Point Cloud and ModelNet40 for the reported experiments [45]. Dataset-specific claims must preserve the official point sampling, splits, augmentation, and metrics.

Training and Evaluation Contract

The compact datasets support fast integration tests, but the reporting contract is the same one required for larger experiments.

Family	Task quantity	Equilibrium quantity	Structural or physical quantity
Fourier	field MSE or relative error	fixed-point residual	elliptic/PDE residual
graph transport	node or graph loss	fixed-point residual	transport residual and relabeling error
homotopy	endpoint task loss	terminal fixed-point residual	trajectory error and evaluation count
distributional	task-head loss	final measure discrepancy	permutation and mask checks

Use a training/validation/test split before fitting normalization statistics. Record the random seed, tensor shapes, solver configuration, equation coefficients, and acceptance thresholds. A small-scale reproduction should be described as such; benchmark equivalence requires the benchmark data and full protocol.

Validation Map

Artifact	What it verifies
tests/test_frontier_data.py	equations, deterministic seeds, masks, batching, gradients, and model integration
tests/test_frontier.py	transition behavior, invariances, solvers, and differentiation
notebooks 17-20	derivations, training loops, diagnostics, plots, and factory construction
scripts/run_notebook_smoke.py	executable release path for every focused lab
scripts/release_audit.py	synchronized publication files, navigation, citations, and plot resolution

Where to Go Next

Question	Page
How do these datasets enter each SILVA transition?	Recent Equilibrium Families
Which classes and builders are public?	Recent Dataset API
How do ordinary dataset adapters work?	Datasets and Preprocessing
Where are the complete citations?	Paper and References