Guarantees and limits
Derivatives
For supported smooth activations, shared polynomial recurrences evaluate activation derivatives without recursive autodiff. Taylor composition yields mathematical derivatives of the represented network through the requested finite order. Floating-point evaluation still introduces rounding error and can suffer cancellation or overflow at high order.
A tower provider reuses an activation evaluation across orders where that activation implements the provider. A custom activation may instead supply only individual derivative fastpaths. Unsupported orders fail explicitly. The PyTorch and JAX jet APIs support affine layers and registered activations; they do not automatically transform an arbitrary framework module into a jet.
Directional jets contain order + 1 coefficient rows. Full mixed jets in
dim dimensions contain binomial(dim + order, order) rows. This output
size remains even when the derivatives are computed without nested autodiff.
The Riccati directional composition path is quadratic in order; the general
composition path is cubic. Benchmark the actual model and requested orders.
Laplacians without the mixed-jet dimension ceiling
The specialized one-hidden-layer operator uses
Delta^k f(x) = sum_h c_h sigma^(2k)(w_h.x + beta_h) ||w_h||^(2k).
It contracts the requested operator directly, avoiding a dense Hessian or a
full order-2k derivative tensor. Computing the preactivations still costs
work proportional to the input dimension. Tanh's individual derivative
fastpath evaluates one tanh and a polynomial of degree n + 1: one activation
evaluation is not constant total arithmetic as n grows.
Deep linear-chain MLPs also have a direct deep_field_laplacian path. It
propagates activation Jacobians and Laplacians without allocating a D x D
identity or enumerating mixed partials. For fixed depth and layer widths, its
work and memory grow linearly with input dimension D. Both backends include
a 5,000-dimensional regression beyond the full mixed-jet allocation budget.
There is no fixed input-dimension cap on this path; available memory and
compute still bound a practical run. Parameter gradients are retained.
Deep repeated Laplacians have three explicit modes:
| Requested operator | Method | Meaning of the result |
|---|---|---|
k = 1 | Forward Laplacian | Exact differentiation, subject to floating-point rounding |
k >= 2, support | Jets over coordinate supports of size at most k | Exact differentiation; work grows with the number of supports |
k >= 2, estimator | Sampled directional derivatives | Unbiased in expectation; a finite run is an estimate |
auto selects support enumeration or sampling according to the budget;
explicitly forcing support bypasses that budget. Use
deep_field_polylaplacian_with_report to inspect the selected method. Its
current concentration report uses the observed range from the first batch
element and output. Treat that report as a diagnostic, not a validated
confidence bound for unseen draws or other outputs. Sampling is not an exact
finite-sample value or an interval-arithmetic proof. Full Hessians,
full mixed jets and other architectures do not inherit the contraction path's
dimension behavior.
Precision and training
Core coefficients are shared across backends. Framework functions, devices, compiler transformations and precision can change rounding; cross-backend identity is a tested property of specific paths, not a universal guarantee. Use matching dtypes and enable JAX 64-bit mode for float64 comparisons.
Spatial jets retain parameter gradients. A differentiable residual does not ensure a well-conditioned optimization problem, good collocation coverage, or convergence to the correct PDE solution. Measure independent validation errors and enforce boundary and initial conditions explicitly.
Certified computations
omnibias.core.verified provides outward-rounded interval and Taylor-model
operations. A certificate applies only to its stated model, domain and
assumptions. A certificate digest establishes integrity; it does not establish
mathematical validity. Checker-earned flags require a successful checker run.
Numerical quadrature and finite-difference approximations must be described as
approximations, with their error bounds when available.