Josh Bongard † & Hod Lipson Computational Synthesis Laboratory Cornell University † Current Address: Department of Computer Science University of Vermont Automated Reverse Engineering of Nonlinear Dynamical Systems July 9, 2007

Model: dθ/dt = ? dω/dt = ?

Best evolved model: dθ/dt = 1.008ω dω/dt = sin(1.0009θ+0/-1.575/-2.673) +f(θ,ω) [f = friction term] Box is flat Rotated -1.57rad Rotated -2.67rad Best human-created model: dθ/dt = ω dω/dt = -9.8Lsin(θ) +f(θ,ω) [L = length of arm] First reported algebraic model describing pendula published by H. Kamerlingh Onnes in Form of these equations still presented to this day in engineering textbooks as the mathematical description of a damped pendulum. Meets criteria (B): Our model matches the best-known model form for a pendulum. Our models contain explicit terms for the friction term f; friction terms are unknown for physical, damped pendula— dependent on mechanical coupling, shape of arm, wind resistance, etc. Our model automatically captures in symbolic form, changes to the system (rotation of the main box).

Best evolved model: dG/dt= ? dA/dt= ? dL/dt= ?

dG/dt= A 2 /(A 2 +1) – 0.01G dA/dt= G( L/(L+1) – A/(A+1) ) dL/dt= -GL/(L+1) Best evolved model: dG/dt= 0.96A 2 /(0.96A 2 +1) dA/dt= G( L/(L+1) – A/(A+1) ) dL/dt= -GL/(L+1)

dG/dt= A 2 /(A 2 +1) – 0.01G dA/dt= G( L/(L+1) – A/(A+1) ) dL/dt= -GL/(L+1) Best evolved model: dG/dt= 0.96A 2 /(0.96A 2 +1) dA/dt= G( L/(L+1) – A/(A+1) ) dL/dt= -GL/(L+1) Lac operon discovered in 1961: Jacob F, Monod J (1961). "Genetic regulatory mechanisms in the synthesis of proteins". J Mol Biol. 3: As of 1998, the best human-created model was: from Keener J, Sneyd J (1998) Mathematical Physiology (Springer). Criteria (B): The result is equal to or better than a result that was accepted as a new scientific result at the time when it was published in a peer-reviewed scientific journal. Took humans 37 years to create this model from observations; Took our algorithm 5 minutes running on a desktop PC to create a very similar model from time series data.

Best evolved model: dH/dt= ? dL/dt= ?

Best evolved model: dH/dt= 3.42x H L dL/dt= 3.10x H L

Best evolved model: dH/dt= 3.42x H L dL/dt= 3.10x H L Best human-created model: dH/dt= αH – βHL dL/dt= -γL + δHL Proposed independently by American biophysicist Alfred Lotka and Italian mathematician Vito Volterra in (Volterra, V. (1926) Jour. du conseil intl. pour l’exp. de la mer) Still forms the basis of most population dynamics models today. Meets criteria (B): Our model is similar, but simpler: No nonlinear terms, yet still conveys: which species in the prey, and which the predator; prey decreases proportionally to amount of predator; predator increases proportionally in response to prey; when integrated, produces coupled oscillations; predator peaks follow prey peaks.

System 1System 2System 3 2 variables dx 1 /dt= -3x 1 x 1 -3x 1 x 2 +2x 2 x 2 dx 2 /dt= -x 1 x 1 -3x 1 x 2 -2x 2 x 2 dx 1 /dt= -3x 1 x 1 +3x 1 x 2 +3x 2 x 2 dx 2 /dt= -3x 1 x 1 -2x 1 x 2 +2x 2 x 2 dx 1 /dt= 3x 1 x 1 -x 1 x 2 -x 2 x 2 dx 2 /dt= x 1 x 1 +3x 1 y 2 -x 2 x 2 Success rate 20% (no partitioning) 100% (partitioning) 0% (no partitioning) 96.7% (partitioning) 0% (no partitioning) 100% (partitioning) 3 variables dx 1 /dt= -3x 1 x 3 -2x 2 x 3 -3x 3 x 3 dx 2 /dt= -3x 1 x 2 +x 1 x 3 -3x 2 x 3 dx 3 /dt= 3x 1 x 2 +3x 1 x 3 –x 2 x 3 dx 1 /dt= -x 1 x 2 +x 1 x 3 -x 2 x 3 dx 2 /dt= x 1 x 1 +2x 1 x 2 +2x 2 x 3 dx 3 /dt= -2x 1 x 1 +x 1 x 2 -3x 2 x 3 dx 1 /dt= -3x 1 x 2 +x 1 x 3 -x 3 x 3 dx 2 /dt= -2x 1 x 3 +3x 2 x 3 +3x 3 x 3 dx 3 /dt= 2x 1 x 2 -2x 1 x 3 -2x 2 x 3 Success rate 0% (no partitioning) 63.3% (partitioning) 0% (no partitioning) 100% (partitioning) 0% (no partitioning) 96.7% (partitioning) 4 variables dx 1 /dt= -x 1 x 1 +2x 2 x 3 +2x 3 x 3 dx 2 /dt= x 1 x 2 -3x 1 x 3 -3x 2 x 3 dx 3 /dt= -x 1 x 1 -x 2 x 4 +3x 4 x 4 dx 4 /dt= -3x 1 x 2 -3x 1 x 4 –3x 3 x 4 dx 1 /dt= x 1 x 4 +x 2 x 4 +x 4 x 4 dx 2 /dt= -3x 1 x 2 -2x 2 x 3 -3x 3 x 4 dx 3 /dt= 2x 1 x 2 -x 1 x 3 +2x 2 x 2 dx 4 /dt= x 1 x 3 +3x 2 x 3 -x 3 x 4 dx 1 /dt= -3x 1 x 1 +3x 1 x 2 +3x 2 x 4 dx 2 /dt= -x 1 x 1 -2x 1 x 3 -3x 4 x 4 dx 3 /dt= -2x 1 x 4 +x 2 x 2 -3x 3 x 4 dx 4 /dt= -x 1 x 2 +2x 1 x 4 -3x 3 x 4 Success rate 0% (no partitioning) 90% (partitioning) 0% (no partitioning) 83.3% (partitioning) 0% (no partitioning) 90% (partitioning) 5 variables dx 1 /dt= -3x 1 x 5 +3x 2 x 3 -3x 2 x 5 dx 2 /dt= -3x 1 x 3 -2x 3 x 4 -x 4 x 5 dx 3 /dt= x 1 x 1 -3x 1 x 4 +x 2 x 4 dx 4 /dt= 3x 1 x 3 -3x 1 x 4 +2x 2 x 2 dx 5 /dt= 3x 1 x 4 +3x 3 x 3 +3x 3 x 4 dx 1 /dt= -2x 2 x 2 +3x 3 x 5 +2x 4 x 5 dx 2 /dt= 3x 1 x 2 +x 1 x 5 -2x 2 x 5 dx 3 /dt= x 1 x 2 +2x 2 x 5 +2x 4 x 5 dx 4 /dt= 2x 1 x 2 +3x 1 x 5 -x 4 x 5 dx 5 /dt= 2x 1 x 5 -x 2 x 5 -2x 5 x 5 dx 1 /dt= 2x 1 x 4 +2x 2 x 3 -x 2 x 4 dx 2 /dt= x 1 x 3 +3x 1 x 4 +x 2 x 4 dx 3 /dt= -2x 1 x 1 +2x 1 x 2 -3x 1 x 3 dx 4 /dt= -3x 2 x 5 +3x 3 x 4 -x 3 x 5 dx 5 /dt= x 1 x 1 +x 1 x 5 +x 2 x 3 Success rate 0% (no partitioning) 76.7% (partitioning) 0% (no partitioning) 76.7% (partitioning) 0% (no partitioning) 76.7% (partitioning) 6 variables dx 1 /dt= -2x 1 x 6 +x 2 x 4 -2x 2 x 6 dx 2 /dt= x 1 x 4 -x 1 x 5 -2x 4 x 4 dx 3 /dt= 2x 2 x 5 -x 3 x 4 +x 5 x 5 dx 4 /dt= -3x 4 x 5 -2x 4 x 6 +2x 5 x 5 dx 5 /dt= x 3 x 6 -2x 4 x 4 -3x 4 x 5 dx 6 /dt= x 3 x 4 -x 3 x 6 +2x 4 x 6 dx 1 /dt= -2x 1 x 3 -3x 2 x 4 +2x 3 x 6 dx 2 /dt= -3x 2 x 4 +x 3 x 4 -x 3 x 6 dx 3 /dt= -x 1 x 2 -x 1 x 3 +x 4 x 6 dx 4 /dt= -x 1 x 4 +x 3 x 5 -2x 4 x 6 dx 5 /dt= 3x 1 x 2 -3x 1 x 6 -x 5 x 5 dx 6 /dt= -3x 1 x 3 -2x 1 x 6 -3x 4 x 6 dx 1 /dt= x 1 x 5 +x 1 x 6 +x 4 x 5 dx 2 /dt= -2x 2 x 5 -2x 2 x 6 +2x 3 x 6 dx 3 /dt= -x 1 x 5 -2x 3 x 4 +x 4 x 4 dx 4 /dt= 3x 1 x 2 +3x 2 x 3 -2x 4 x 5 dx 5 /dt= -3x 1 x 5 +x 2 x 2 +3x 2 x 6 dx 6 /dt= -x 2 x 5 -2x 3 x 5 -3x 5 x 6 Success rate 0% (no partitioning) 80% (partitioning) 0% (no partitioning) 70% (partitioning) 0% (no partitioning) 93% (partitioning) Criteria (G): The result solves a problem of indisputable difficulty in its field. Algorithm presented with time series data from nonlinear, coupled systems of increasing size. Algorithm correctly inferred the underlying form in 5 minutes. 2 variables Best-known predator-prey model; single pendulum model 3 variables Best-known lac operon model of E. coli as of variables Best-known lac operon model of E. coli as of 2003 (Yildirim & Mackey, 2003)

Why should this entry be considered “best”? Work appeared in one of the world’s most prestigious scientific journals, and not in an evolutionary computation journal: Was favorably reviewed by four renowned scientists outside of the EC community. One reviewer remarked that this approach “…has a chance of changing the way some disciplines conduct science”. Citation—Bongard J. and Lipson H.(2007). Automated reverse engineering of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 104(24):