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
ε
there exist a smooth solution
< ε ( x )
[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.
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