Adaptive differential equations

Use this page when a differential equation needs variable step sizes and you must interpret what local error control does, and does not, say about the solution.

The scientific question¶

An adaptive integrator allocates more steps where the solution is difficult and fewer where it is smooth. The scientific question is whether a local numerical error estimate, measured against meaningful component scales, is an adequate control signal for the observable and time interval of interest.

Local error control limits an estimate attached to one proposed step. Global solution error accumulates and propagates across accepted steps. It depends on stability, stiffness, events, tolerances, precision, and the dynamics, so it is not equal to the requested local tolerance.

Mathematical objects¶

Consider $d\mathbf{y}/dt=\mathbf{f}(t,\mathbf{y},\boldsymbol{\theta})$ . An embedded or paired method proposes a state and an error estimate $\widehat{\mathbf{e}}$ . Each component is compared with a scale that combines an absolute floor and relative magnitude. The scaled norm, controller order, safety factor, growth limits, and rejection history are part of the executed algorithm.

Component units matter. One shared absolute tolerance can be meaningless when state leaves represent different physical dimensions or very different scientific scales.

Core derivation¶

For a scalar component, define a tolerance scale and normalized error. A common controller pattern is:

\begin{aligned} s &= \mathrm{atol}+\mathrm{rtol}|y|, \\ E &= \left\|\widehat{\mathbf{e}}/\mathbf{s}\right\|, \\ \mathrm{accept} &\iff E\leq 1, \\ h_{\mathrm{next}} &=h\,\mathrm{clip}\left( \gamma E^{-1/(p+1)},q_{\min},q_{\max} \right). \end{aligned}

(1)

In (1), $p$ identifies the error model order, $\gamma$ is a safety factor, and the clip prevents extreme step changes. An implementation must define the norm, the zero-error branch, what state enters $\mathbf{s}$ , and whether rejected proposals affect diagnostics or adjoints.

What the ecosystem already owns¶

Diffrax is the runtime owner for adaptive differential-equation solving in this ecosystem. Solver formulas, controllers, dense output, events, adjoint strategies, result states, and transform behavior belong to Diffrax. Jaxstro’s fixed-step ODE helpers remain a smaller, separate contract; they are not an adaptive-solver framework.

What Jaxstro may add¶

A future adapter may translate unit-bearing state into an explicit per-leaf tolerance policy, record solver and controller settings, and connect convergence studies to provenance. It may also define evidence envelopes for particular scientific observables. No adaptive Jaxstro adapter exists today.

Any adapter must expose Diffrax ownership and return its statuses rather than masking them behind a success boolean.

Evidence required before implementation¶

Required evidence would include:

analytic and high-accuracy reference solutions across nonstiff and stiff examples relevant to the claimed scope;
tolerance sweeps for state error and downstream observables;
unit-rescaling parity for heterogeneous state leaves;
event-time and discontinuity tests with explicitly bounded claims;
adjoint comparisons against independent finite differences on smooth cases;
failure cases for step underflow, nonfinite states, and exhausted steps; and
deterministic provenance for solver, controller, tolerances, and statuses.

Claim boundary¶

This page does not claim that one solver or adjoint is best, does not report performance, and does not add adaptive integration to Jaxstro.

Connected foundations and methods¶

Use Functions, units, and scales for scales and units, and From mathematical relations to differentiable programs for the distinction between a mathematical relation and an executed program. Compare the current fixed-step methods in Fixed-step ODE integration, derivative contracts in Autodiff products, and cumulative integral mechanics in Sampled Newton-Cotes integration.