Energy-conditioned mean-reverting SSM

Integrate this into a stochastic recurrent layer or diagonal state-space model, rather than adding only a loss penalty. For a batch of sequences, let h_k in R^d be the hidden state, x_k the input, gamma_0>0 the base decay, sigma the injected-noise scale, B the desired mean squared state norm per timestep, and lambda the nonnegative feedback multiplier. Use an Euler discretization:

inputs: x[0:T], state h=0, energy=0, lambda=lambda0
for k in 0,...,T-1:
    gamma = sqrt(gamma0^2 + 2*sigma^2*lambda)
    noise = sigma * sqrt(dt) * Normal(0,I)
    h = h + dt*(-gamma*h + input_projection(x[k])) + noise
    energy = energy + mean(h*h)
    lambda = max(0, lambda + eta_lambda*(energy/(k+1) - B))
    y[k] = output_projection(h)

Use stop-gradient through the controller update initially, and clip lambda to a fixed maximum to prevent over-contraction. The paper-derived computation is the square-root relation between the quadratic path-tilt multiplier and effective mean reversion. Estimate energy empirically as the batch mean of h_k squared, and tune B from the baseline model's median per-token state energy; tune eta_lambda over a logarithmic grid. Update lambda per truncated-BPTT window instead of every token if token-level feedback is noisy. First test a 1–4 layer stochastic diagonal SSM or GRU on sequential MNIST and a small character-level language model. Compare fixed-decay dynamics, an explicit activation penalty alpha times the sum of squared states, and the adaptive drift at matched parameter count and noise level. Record validation loss or perplexity, hidden-state energy variance, exploding-gradient frequency, and performance on sequences longer than those seen in training. Success means a narrower stable energy distribution and equal-or-better validation performance without a large explicit penalty; failure would be systematically worse accuracy from excessive contraction or materially slower optimization.