Universal differential equation

This article is about an algebraic differential equation with an universality property. For universal differential equation in physics-informed machine learning, see Neural differential equation.

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 ${\displaystyle P(y',y'',y''',...,y^{(n)})=0}$ is a UDE if for any continuous real-valued function ${\displaystyle f}$ and for any positive continuous function ${\displaystyle \varepsilon }$ there exist a smooth solution ${\displaystyle y}$ of ${\displaystyle P(y',y'',y''',...,y^{(n)})=0}$ with ${\displaystyle |y(x)-f(x)|<\varepsilon (x)}$ for all ${\displaystyle x\in \mathbb {R} }$.^[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]

Examples

• Rubel found the first known UDE in 1981. It is given by the following implicit differential equation of fourth-order:^[1][2] ${\displaystyle 3y^{\prime 4}y^{\prime \prime }y^{\prime \prime \prime \prime 2}-4y^{\prime 4}y^{\prime \prime \prime 2}y^{\prime \prime \prime \prime }+6y^{\prime 3}y^{\prime \prime 2}y^{\prime \prime \prime }y^{\prime \prime \prime \prime }+24y^{\prime 2}y^{\prime \prime 4}y^{\prime \prime \prime \prime }-12y^{\prime 3}y^{\prime \prime }y^{\prime \prime \prime 3}-29y^{\prime 2}y^{\prime \prime 3}y^{\prime \prime \prime 2}+12y^{\prime \prime 7}=0}$
• Duffin obtained a family of UDEs given by:^[3]

${\displaystyle n^{2}y^{\prime \prime \prime \prime }y^{\prime 2}+3n(1-n)y^{\prime \prime \prime }y^{\prime \prime }y^{\prime }+\left(2n^{2}-3n+1\right)y^{\prime \prime 3}=0}$ and ${\displaystyle ny^{\prime \prime \prime \prime }y^{\prime 2}+(2-3n)y^{\prime \prime \prime }y^{\prime \prime }y^{\prime }+2(n-1)y^{\prime \prime 3}=0}$, whose solutions are of class ${\displaystyle C^{n}}$ for n > 3.

• Briggs proposed another family of UDEs whose construction is based on Jacobi elliptic functions:^[4]

${\displaystyle y^{\prime \prime \prime \prime }y^{\prime 2}-3y^{\prime \prime \prime \prime }y^{\prime \prime }y^{\prime }+2\left(1-n^{-2}\right)y^{\prime \prime 3}=0}$, where n > 3.

• Bournez and Pouly proved the existence of a fixed polynomial vector field p such that for any f and ε there exists some initial condition of the differential equation y' = p(y) that yields a unique and analytic solution satisfying |y(x) − f(x)| < ε(x) for all x in R.^[2]

See also

• Zeta function universality
• Hölder's theorem