A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line to any desired level of accuracy.
Precisely, a (possibly implicit) differential equation is a UDE if for any continuous real-valued function and for any positive continuous function there exist a smooth solution of with for all .[1]
The existence of an UDE has been initially regarded as an analogue of the universal Turing machine for analog computers, because of a result of Shannon that identifies the outputs of the general purpose analog computer with the solutions of algebraic differential equations.[1] However, in contrast to universal Turing machines, UDEs do not dictate the evolution of a system, but rather sets out certain conditions that any evolution must fulfill.[2]
^ abRubel, Lee A. (1981). "A universal differential equation". Bulletin of the American Mathematical Society. 4 (3): 345–349. doi:10.1090/S0273-0979-1981-14910-7. ISSN 0273-0979.
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