Solving Differential Equations in Julia II

There are two complaints about a solver that are constantly mistaken for one another: that it is slow, and that it cannot state your problem at all. The first is an argument about hardware. The second is a structural fact, and only the second is worth a book.

This is the second of two volumes. Volume I took the axis of structure — the deterministic equation the standard interface cannot state: mass matrices, high-index differential-algebraic systems, the symbolic-numeric layer. This volume takes up what remains: the equation that is stochastic, delayed, discontinuous, or unknown. Stochastic differential equations whose solution is a distribution; delay equations whose initial condition is a function; jumps and hybrid dynamics; continuation past the fold; performance and GPU-scale ensembles; the differentiation of the solver itself; universal differential equations that recover physics from data; and a verification discipline that ends in a machine-checked proof. Together they map where the Julia SciML stack states what Python cannot.

Every comparison gives the rival its best configuration — and this volume concedes a great deal, scoring the differentiable challenger as the genuine rival it is wherever it competes. Every timing carries its tolerance. Every claim rests on a work-precision diagram, and every solution is verified against a manufactured one. Slow is an inconvenience; cannot is a theorem. These books keep them apart.