Build a neural ODE or invertible transformation whose primitive layers are flows of learned gradient vector fields, then synthesize non-gradient directions using short Lie-bracket commutator products. The paper's bounded-bracket-generation result predicts that restricted gradient primitives can approximate a much larger class of diffeomorphisms than a plain stack of gradient flows.
Represent every nonnegative equal-mass one-dimensional state by its CDT quantile map relative to a fixed reference density, then train the neural dynamics model in this transformed space rather than on Eulerian grid values. The latent manifold for translations and transport-dominated evolution is substantially flatter: linear transport lies in the span of the initial transformed state and the constant function, while nonlinear conservative dynamics have algebraic approximation error bounds.
Approximate the exact event-by-event lifted sampler by drawing independent Poisson jump counts over a short interval and applying compatible discrete moves in parallel. This converts sequential neighbor events into batched GPU-friendly updates while retaining the Hamiltonian rate structure; the step size controls the error-versus-throughput tradeoff.
When a neural model defines a product distribution over coordinates, compute the discrepancy to a target product distribution from per-coordinate divergences using the exact tensorization rule instead of sampling full vectors and estimating a joint divergence. Implement KL, chi-squared, and squared-Hellinger variants as drop-in losses, with an optional learned choice among these mathematically tensorizable families.
Place doubly stochastic stream mixing immediately before a quantizer, activation compressor, or latent bottleneck and jointly optimize task loss with estimated code length. The paper's entropy argument says that this linear mixing cannot increase differential entropy, so it can provide cross-stream representation capacity without an ideal entropy-rate penalty; the entropy bottleneck then learns which feature values deserve bits.
Replace an unconstrained q-way polynomial or tensorized feature layer with separate decomposable and primitive interaction channels. The decomposable channel models interactions explainable as products of lower physical-weight feature blocks, while the primitive channel captures residual factors that cannot be represented by those products. This should reduce redundant high-order parameters and provide a controllable inductive bias for compositional or disentangled representations.
Replace backpropagated gradients in a Langevin sampler with a simultaneous-perturbation estimate obtained from two noisy evaluations of a neural energy or simulator-defined negative log-density. This enables posterior or latent-space sampling when the energy contains nondifferentiable code, stochastic simulation, discrete operations, or an inaccessible neural-network gradient.
Use the measured power-exponential covering dimension of an activation manifold to choose the growth rate of latent codebooks, prototypes, or quantization resolution. The goal is to avoid spending parameters on a representation whose attainable resolution exceeds the intrinsic covering complexity.
Replace a complex latent vector x in C^d by squared magnitudes of m learned complex linear projections. Set m equal to 2d: the paper proves that m less than or equal to 2d minus 1 cannot generically preserve the latent up to global phase, whereas m equal to 2d is generically sufficient, giving a principled minimal width for a phase-invariant neural bottleneck.
Use the paper's one-bit compressed-sensing lower bound to choose the number of binary latent measurements and to set a nonzero achievable-error floor during training. A sign bottleneck should not be given an unrealistically small bit budget: for approximately sparse latents, the target reconstruction error scales no faster than a power of effective sparsity divided by the number of sign measurements.