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

References

1. Rubel, 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.
2. Pouly, Amaury; Bournez, Olivier (2020-02-28). "A Universal Ordinary Differential Equation". Logical Methods in Computer Science. 16 (1). arXiv: 1702.08328 . doi:10.23638/LMCS-16(1:28)2020. S2CID   4736209.
3. Duffin, R. J. (1981). "Rubel's universal differential equation". Proceedings of the National Academy of Sciences. 78 (8): 4661–4662. Bibcode:1981PNAS...78.4661D. doi: 10.1073/pnas.78.8.4661 . ISSN   0027-8424. PMC   320216 . PMID   16593068.
4. Briggs, Keith (2002-11-08). "Another universal differential equation". arXiv: math/0211142 .

External links

• Wolfram Mathworld page on UDEs