THE DIFFERENTIAL EQUATION THAT SOLVES EVERY PROBLEM OR HOW TO LIE - PowerPoint PPT Presentation

THE DIFFERENTIAL EQUATION THAT SOLVES EVERY PROBLEM OR HOW TO LIE WITH UNIVERSAL EQUATIONS I IV II Tamás Kalmár-Nagy III Budapest University of Technology and Economics Balázs Sándor Department of Hydraulic and Water Resources Engineering Budapest University of Technology and Economics Griffith University, School of Engineering, Gold Coast, Australia

2 SOLVING EVERY PROBLEM 153 Miklós Farkas Seminar on Applied Analysis, 2018/02/15

3 THANK YOU! 153 QUESTIONS? Miklós Farkas Seminar on Applied Analysis, 2018/02/15

4 SOLVING EVERY PROBLEM 153 Theorem (Rubel, 1981) : There exists a fourth- order Algebraic Differential Equation (ADE) P  y  , y  , y  , y    0 that has a solution arbitrarily close to any smooth function for all t Def: Let F be a class of real-valued functions defined on D ⊆ R. An ADE is called universal with respect to F(D), if every continuous function on F can be uniformly approximated by F-solutions of this ADE on D. Miklós Farkas Seminar on Applied Analysis, 2018/02/15

5 CONSTRUCTING UNIVERSAL EQUATIONS 153 Universal Equation Recipe: 1. Take an ounce of sigmoid 2. Differentiate, then differentiate again 3. Eliminate constants 4. Shake, not stir Miklós Farkas Seminar on Applied Analysis, 2018/02/15

6 1. TAKE AN OUNCE OF SIGMOID 153 finite sigmoid f on [0,1] f  0   0, f  1   1 # f   0   f   1   0 # f   t   0, t   0,1  . # y  t   Af   t     B The constants A, B, α, β serve to shift and scale the sigmoid Miklós Farkas Seminar on Applied Analysis, 2018/02/15

7 WHY A SIGMOID? 153 f is a sigmoid on [0,1] Any continuous function can be arbitrarily approximated by concatenating scaled and shifted sigmoids y  t   Af   t     B Miklós Farkas Seminar on Applied Analysis, 2018/02/15

8 2. DIFFERENTIATE, THEN DIFFERENTIATE AGAIN 153 The constant B can simply be eliminated by differentiating y   t   A  f    t    More constants? More differentiation… y   t   A  2 f     t    ,  y  n   t   A  n f  n    t    . # Miklós Farkas Seminar on Applied Analysis, 2018/02/15

9 3. ELIMINATE CONSTANTS 153 Look for equations of the form P  y  ,  , y  n    P  A  f  ,  , A  n f  n    0 that are independent of the constants A and α. Introduce the “kernel” g  t   f    t    P  A  g , A  2 g  , A  3 g  , A  4 g  ,    0 Miklós Farkas Seminar on Applied Analysis, 2018/02/15

10 4. SHAKE, NOT STIR (RUBEL’S EXAMPLE) 153 Assume a given form for the kernel 1  1  t  1 e t 2  1 g  t   otherwise 0 P  A  g , A  2 g  , A  3 g  , A  4 g    0 Shake with some algebra 3 y  4 y  y  2  4 y  4 y  2 y   6 y  3 y  2 y  y   24 y  2 y  4 y   12 y  3 y  y  3  29 y  2 y  3 y  2  12 y  7  0. # Miklós Farkas Seminar on Applied Analysis, 2018/02/15

11 OTHER UNIVERSAL EQUATIONS 153 g  t    1  t 2  n Duffin 1981  2 n 2  3 n  1  y  3  3  n  1  ny  y  y   n 2 y  2 y   0 y  y  2  3 y  y  y   2  1  n  y  3  0 1 Briggs 2002 Miklós Farkas Seminar on Applied Analysis, 2018/02/15

12 CAN WE CONSTRUCT ALL UNIVERSAL EQUATIONS? 153 To "eliminate" the constants A and α we need P be a homogeneous polynomial of degree D P  A  g , A  2 g  , A  3 g  , A  4 g  ,     A  g  D P  g  g ,  2 g  g ,  3 g  g ,   0 g   t  h  t   Introduce the logarithmic derivative g  t  g   hg , # g   h  g  hg    h   h 2  g , # g    h   3 hh   h 3  g , # g  4    h   4 hh   3 h  2  6 h 2 h   h 4  g # P   h ,  2  h   h 2  ,  3  h   3 hh   h 3  ,  4  h   4 hh   3 h  2  6 h 2 h   h 4  ,    #  c i  b 1  2 b 2  3 b 3  4 b 4 h b 1  h   h 2  b 2  h   3 hh   h 3  b 3  h   4 hh   3 h  2  6 h 2 h   h 4  b 4  0. # i b 1  2 b 2  3 b 3  4 b 4  W weight Miklós Farkas Seminar on Applied Analysis, 2018/02/15

13 CONSTRUCTING UNIVERSAL EQUATIONS 153 The derivatives of h in terms of the derivatives of y h  y  h   h 2  y  y  , y  , h   y  y    y   2 , h   3 hh   h 3  y  4  y  2 y  , h   2  y   3  3 y  y  y   y  2 y  # , y  3 h   4 hh   3 h  2  6 h 2 h   h 4  y  5  h    6  y   4  12 y   y   2 y   3 y  2  y   2  4 y  2 y  y  4   y  3 y  5  y  , . y  4 b 1 b 2 b 3 b 4  0  y  y  y  4  y  5  c i y  y  y  y  i b 1  2 b 2  3 b 3  4 b 4  W Miklós Farkas Seminar on Applied Analysis, 2018/02/15

14 CONSTRUCTING UNIVERSAL EQUATIONS 153 W  2 b 1  2 b 2  2 b 1 b 2 W   b i 0 1 1 2 0 0 W   b i  y   b 1  y   b 2  c 1 y  y   c 2 y  2  c i  y   i , b 1  2 b 2  W The simplest Universal Differential Equation? Miklós Farkas Seminar on Applied Analysis, 2018/02/15

15 CONSTRUCTING UNIVERSAL EQUATIONS 153 W  3 b 1  2 b 2  3 b 3  3 W   b i b 1 b 2 b 3 3 0 0 0 1 1 1 0 2 0 0 1 W   b i  y   b 1  y   b 2  y   b 3  c 1 y  3  c 2 y  y  y   c 3 y  2 y   c i  y   i , b 1  2 b 2  3 b 3  W Miklós Farkas Seminar on Applied Analysis, 2018/02/15

16 CONSTRUCTING UNIVERSAL EQUATIONS 153 b 1  2 b 2  3 b 3  4 b 4  5 b 5  5 W  5 b 1 b 2 b 3 b 4 b 5 W   b i 5 0 0 0 0 0 3 1 0 0 0 1 1 2 0 0 0 2 2 0 1 0 0 2 0 1 1 0 0 3 1 0 0 1 0 3 0 0 0 0 1 4 W   b i  y   b 1  y   b 2  y   b 3  y  5    b 5  b 4  y  6   c i  y   # i , b 1  2 b 2  3 b 3  4 b 4  5 b 5  W c 1 y  5  c 2 y  y  3 y   c 3 y  2 y  2 y  4   c 4 y  2 y  y  2  c 5 y  3 y  y  5   c 6 y  3 y  y  4   c 7 y  4 y  6  # WHAT ARE THE CORRESPONDING KERNELS/SIGMOIDS? Miklós Farkas Seminar on Applied Analysis, 2018/02/15

17 ANOTHER DIRECTION: LIE GROUPS 153 The theory of Lie-groups contains powerful tools to analysing differential equations. It has been developed by Marius Sophus Lie at the end of the XIX. -th century. Miklós Farkas Seminar on Applied Analysis, 2018/02/15

LIE-POINT TRANSFORMATION GROUPS AS SYMMETRY 18 GROUPS OF DIFFERENTIAL EQUATIONS 153 Let G be a Lie-group acting on the M manifold. Let M be locally the total space of the independent ( m ) and dependent ( n ) � = � ≃ ℝ � × ℝ � variables of a differential equation. G is a symmetry group of the differential equation if it transforms its solutions into another solutions on M . Example 1. : Let G be the one parameter group of the shifts of the independent variable ( t ) of an ODE (with y(t) as unknown). The group elements act on the (t,y) plane as: � ∗ = � + ε, � ∗ = � → � � �, �, ε = � + ε, � � �, �, ε = � � ∈ �, �, � ∈ � ≃ ℝ × ℝ Miklós Farkas Seminar on Applied Analysis, 2018/02/15

AFFINE GROUP OF THE SOLUTION PLANE OF UNIVERSAL 19 DIFFERENTIAL EQUATIONS 153 f  t   Af   t     B To be universal, a differential equation must have a solution with which an arbitrary function can be uniformly approximated. Hence a universal differential equation must be form invariant under translations and scaling. Working definition An algebraic differential equation is universal if i.) it has a nonconstant solution whose derivatives vanish at the endpoints of a finite interval, ii.) it has the four parameter symmetry group of the affine transformations of the plane of the independent and dependent variable. Miklós Farkas Seminar on Applied Analysis, 2018/02/15