Bayesian, Joint, Dynamic, and Certified Equilibria

Four additional families extend the same SILVA state contract into uncertainty, coupled input optimization, long-horizon dynamics, and certified output bounds. They are implementations inside SILVA rather than separate execution systems.

The common decomposition remains

\[ z^\star=\sigma\!\left(S_\theta(x)+H_\theta(z^\star) +L_\theta(z^\star)+G_\theta(z^\star)\right), \qquad \widehat y=Q_\psi(z^\star). \]

What changes is the meaning of the equilibrium state and the constraints on its transition. The package keeps every transition, readout, solver, and scale choice replaceable.

1. Bayesian Equilibrium

The Bayesian family places a diagonal Gaussian posterior over transition parameters:

\[ q_\phi(\theta)=\mathcal N(\mu,\operatorname{diag}(\sigma^2)), \qquad \theta_s=\mu+\sigma\odot\epsilon_s, \qquad \epsilon_s\sim\mathcal N(0,I). \]

Each sample defines its own fixed point,

\[ z_s^\star=T_{\theta_s}(z_s^\star,x), \qquad \widehat y_s=Q_\psi(z_s^\star). \]

The predictive moments are

\[ \bar y=\frac{1}{S}\sum_{s=1}^S\widehat y_s, \qquad \widehat{\operatorname{Var}}(y\mid x) =\frac{1}{S}\sum_{s=1}^S(\widehat y_s-\bar y)^2. \]

SILVABayesianAffineTransition row-normalizes every sampled state matrix so its infinity norm remains below state_scale. SILVABayesianDEQ can solve samples independently or warm-start sample \(s+1\) from sample \(s\). The latter captures the sequential-inference mechanism described by Gao et al. [94].

from silva_networks import SILVABayesianDEQ

model = SILVABayesianDEQ(
    input_dim=32,
    state_dim=128,
    output_dim=10,
    posterior_samples=8,
    sequential=True,
)
result = model(features, seed=17, return_result=True)
loss = task_loss(result.output, labels) + beta * model.kl_divergence()

For a different posterior, implement sample_parameters, a shape-preserving forward(state, inputs, sample), and optionally kl_divergence. The solver and predictive aggregation do not need to change.

2. Joint Representation and Input Inference

Joint inference turns the optimized input into part of the equilibrium state. For observation \(y\), representation \(z\), and optimized input \(u\),

\[ \begin{aligned} z^\star &=T_\theta(z^\star,u^\star,y),\\ u^\star &=P_{\mathcal C}\!\left(u^\star -\eta g_\phi(u^\star,z^\star,y)\right). \end{aligned} \]

Packing \(w=(z,u)\) gives one fixed-point equation \(w^\star=F(w^\star;y)\). This removes the outer sequence of complete forward/backward passes used by a naive input optimizer. The construction follows the joint-inference mechanism of Gurumurthy et al. [95].

from silva_networks import SILVAJointInferenceEquilibrium

model = SILVAJointInferenceEquilibrium(
    observation_dim=64,
    state_dim=256,
    optimized_input_dim=128,
    output_dim=64,
    representation_transition=my_transition,
    input_update=my_projected_update,
    readout=my_decoder,
)
result = model(observation, return_result=True)

The default input branch is a projected quadratic update. Replace it with a latent reconstruction step, adversarial projection, proximal operator, or task-adaptation update. Both branches must preserve their declared state shape.

3. Implicit Spatiotemporal Dynamics

Let \(F=F_{\rm known}+F_{\rm learned}\). A theta-method step solves

\[ u_{n+1}=u_n+\Delta t\left[(1-\vartheta)F(u_n,c_n) +\vartheta F(u_{n+1},c_n)\right]. \]

The corresponding root is

\[ R_{n+1}(v)=v-u_n-\Delta t\left[(1-\vartheta)F(u_n,c_n) +\vartheta F(v,c_n)\right]=0. \]

For \(\vartheta=1\) this is backward Euler; \(\vartheta=\tfrac12\) is the trapezoidal rule. Every time step is a SILVA equilibrium, while the complete trajectory remains differentiable. This captures the implicit-step mechanism used for stable long-horizon hybrid dynamics [96].

from silva_networks import (
    SILVAImplicitSpatiotemporalEquilibrium,
    SILVAPeriodicDiffusion1D,
)

model = SILVAImplicitSpatiotemporalEquilibrium(
    known_dynamics=SILVAPeriodicDiffusion1D(diffusivity=0.05),
    learned_dynamics=closure_model,
    projector=boundary_projector,
    dt=0.01,
    theta=1.0,
    steps=200,
)
trajectory = model(initial_state)

The spatial operator can be a convolution, graph discretization, Fourier operator, finite-volume residual, U-Net, or a composition of named SILVA branches. projector is where boundary conditions, conservation corrections, or admissible-state constraints belong.

4. Certified Contractive Equilibrium

The certified family uses

\[ z^\star=\phi(Wz^\star+Ux+b), \qquad \lVert W\rVert_\infty<1, \]

with monotone activation \(\phi\). For an input box \([\underline x,\overline x]\), split every matrix into \(A=A^++A^-\), where \(A^+=\max(A,0)\) and \(A^-=\min(A,0)\). Signed affine bounds obey

\[ \underline{Av}=A^+\underline v+A^-\overline v, \qquad \overline{Av}=A^+\overline v+A^-\underline v. \]

The lower and upper states are solved jointly as another fixed point. Their readout intervals certify class \(c\) whenever

\[ \underline y_c-\max_{j\ne c}\overline y_j>0. \]

This is the interval-equilibrium idea behind IBP-MonDEQ [97]. The exported affine ReLU system can also be passed to semialgebraic certification programs [98].

from silva_networks import SILVACertifiedEquilibrium

model = SILVACertifiedEquilibrium(
    input_dim=784,
    state_dim=256,
    output_dim=10,
    activation="relu",
    contraction=0.8,
)
logits = model(images.flatten(1))
certificate = model.certify(images.flatten(1), radius=0.05, labels=labels)
system = model.semialgebraic_system()

The interval routine is sound for the implemented contractive affine system. It is not a substitute for the complete monotone-operator parameterization or external semidefinite program used by a different certificate. Those can be added through a new transition and certificate backend while retaining the same SILVA input/state/readout boundary.

5. Scale Without Changing the Question

Every family has three generated execution tiers:

from silva_networks import silva_family_experiment_protocol

protocol = silva_family_experiment_protocol("silva_implicit_spatiotemporal")
for tier in protocol.tiers:
    print(tier.tier, tier.dataset.name, tier.resources.storage)

Materialize the complete run input before downloading or training:

python experiments/reproduction/run_family_protocol.py \
  --family silva_implicit_spatiotemporal \
  --tier workstation \
  --work-dir runs/spatiotemporal/workstation

The smoke tier uses generated or bundled data. The workstation tier validates the source loader, checkpoint/resume path, metric, and memory on a recorded subset. The full tier removes the sample cap and restores the cited split, resolution, horizon, training budget, and evaluator. PDE families can use the PDEBench route [93].

Where to Go Next

Question	Page
How is evidence promoted across scales?	Evidence and Source-Scale Experiments
Where are all family-specific routes?	Family Reproduction Dossiers
Which classes expose these mechanisms?	Advanced Expansion API
Which notebooks execute each derivation?	Notebooks