Research ArticleCOMPUTER SCIENCE

# AI Feynman: A physics-inspired method for symbolic regression

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Vol. 6, no. 16, eaay2631

### Tables

• Table 1 Functions optionally included in brute-force search.

The following three subsets are tried in turn: “+−*/><~SPLICER”, “+−*/> 0~” and “+−*/><~REPLICANTS0”.

 Symbol Meaning Arguments + Add 2 * Multiply 2 − Subtract 2 / Divide 2 > Increment 1 < Decrement 1 ∼ Negate 1 0 0 0 1 1 0 R sqrt 1 E exp 1 P π 0 L ln 1 I invert 1 C cos 1 A abs 1 N arcsin 1 T arctan 1 S sin 1
• Table 2 Hyperparameters in our algorithm and the setting we use in this paper.

 Symbol Meaning Setting ϵbr Tolerance in brute-forcemodule 10−5 ϵpol Tolerance in polynomialfit module 10−4 ϵNN0 Validation errortolerance for neuralnetwork use 10−2 ϵsep Tolerance forseparability 10 ϵNN ϵsym Tolerance for symmetry 7 ϵNN ϵbfsep Tolerance in brute-forcemodule afterseparability 10 ϵNN ϵpolsep Tolerance in polynomialfit module afterseparability 10 ϵNN λ Importance of accuracyrelative to complexity Nd1/2
• Table 3 Unit table used for our automated dimensional analysis.

 Variables Units m s kg T V a, g Acceleration 1 −2 0 0 0 h, ℏ, L, Jz Angular momentum 2 −1 1 0 0 A Area 2 0 0 0 0 kb Boltzmann constant 2 −2 1 −1 0 C Capacitance 2 −2 1 0 −2 q, q1, q2 Charge 2 −2 1 0 −1 j Current density 0 −3 1 0 −1 I, I0 Current Intensity 2 −3 1 0 −1 ρ, ρ0 Density −3 0 1 0 0 θ, θ1, θ2, σ, n Dimensionless 0 0 0 0 0 g_, kf, γ, χ, β, α Dimensionless 0 0 0 0 0 pγ, n0, δ, f, μ Dimensionless 0 0 0 0 0 n0, δ, f, μ, Z1, Z2 Dimensionless 0 0 0 0 0 D Diffusion coefficient 2 −1 0 0 0 μdrift Drift velocity constant 0 −1 1 0 0 pd Electric dipole moment 3 −2 1 0 −1 Ef Electric field −1 0 0 0 1 ϵ Electric permitivity 1 −2 1 0 −2 E, K, U Energy 2 −2 1 0 0 Eden Energy density −1 −2 1 0 0 FE Energy flux 0 −3 1 0 0 F, Nn Force 1 −2 1 0 0 ω, ω0 Frequency 0 −1 0 0 0 kG Grav. coupling (Gm1m2) 3 −2 1 0 0 H Hubble constant 0 −1 0 0 0 Lind Inductance −2 4 −1 0 2 nrho Inverse volume −3 0 0 0 0 x, x1, x2, x3 Length 1 0 0 0 0 y, y1, y2, y3 Length 1 0 0 0 0 z, z1, z2, r, r1, r2 Length 1 0 0 0 0 λ, d1, d2, d, ff, af Length 1 0 0 0 0 I1, I2, I*, I*0 Light intensity 0 −3 1 0 0 B, Bx, By, Bz Magnetic field −2 1 0 0 1 μm Magnetic moment 4 −3 1 0 −1 M Magnetization 1 −3 1 0 −1 m, m0, m1, m2 Mass 0 0 1 0 0 μe Mobility 0 1 −1 0 0 p Momentum 1 −1 1 0 0 G Newton’s constant 3 −2 −1 0 0 P* Polarization 0 −2 1 0 −1 P Power 2 −3 1 0 0 pF Pressure −1 −2 1 0 0 R Resistance −2 3 −1 0 2 μS Shear modulus −1 −2 1 0 0 Lrad Spectral radiance 0 −2 1 0 0 kspring Spring constant 0 −2 1 0 0 σden Surface charge density 0 −2 1 0 −1 T, T1, T2 Temperature 0 0 0 1 0 κ Thermal conductivity 1 −3 1 −1 0 t, t1 Time 0 1 0 0 0 τ Torque 2 −2 1 0 0 Avec Vector potential −1 1 0 0 1 u, v, v1, c, w Velocity 1 −1 0 0 0 V, V1, V2 Volume 3 0 0 0 0 ρc, ρc0 Volume charge density −1 −2 1 0 −1 Ve Voltage 0 0 0 0 1 k Wave number −1 0 0 0 0 Y Young modulus −1 −2 1 0 0
• Table 4 Tested Feynman equations, part 1.

Abbreviations in the “Methods used” column: da, dimensional analysis; bf, brute force; pf, polyfit; ev, set two variables equal; sym, symmetry; sep, separability. Suffixes denote the type of symmetry or separability (sym–, translational symmetry; sep*, multiplicative separability; etc.) or the preprocessing before brute force (e.g., bf-inverse means inverting the mystery function before bf).

 Feynman Eq. Equation Solution Time (s) Methods Used Data Needed Solved By Eureqa Solved W/oda NoiseTolerance I.6.20a f=e−θ2/2/2π 16 bf 10 No Yes 10−2 I.6.20 f=e−θ22σ2/2πσ2 2992 ev, bf-log 102 No Yes 10−4 I.6.20b f=e−(θ−θ1)22σ2/2πσ2 4792 sym–, ev, bf-log 103 No Yes 10−4 I.8.14 d=(x2−x1)2+(y2−y1)2 544 da, pf-squared 102 No Yes 10−4 I.9.18 F=Gm1m2(x2−x1)2+(y2−y1)2+(z2−z1)2 5975 da, sym–, sym–, sep∗, pf-inv 106 No Yes 10−5 I.10.7 m=m01−v2c2 14 da, bf 10 No Yes 10−4 I.11.19 A = x1y1 + x2y2 + x3y3 184 da, pf 102 Yes Yes 10−3 I.12.1 F = μNn 12 da, bf 10 Yes Yes 10−3 I.12.2 F=q1q24πϵr2 17 da, bf 10 Yes Yes 10−2 I.12.4 Ef=q14πϵr2 12 da 10 Yes Yes 10−2 I.12.5 F = q2Ef 8 da 10 Yes Yes 10−2 I.12.11 F = q(Ef + Bv sin θ) 19 da, bf 10 Yes Yes 10−3 I.13.4 K=12m(v2+u2+w2) 22 da, bf 10 Yes Yes 10−4 I.13.12 U=Gm1m2(1r2−1r1) 20 da, bf 10 Yes Yes 10−4 I.14.3 U = mgz 12 da 10 Yes Yes 10−2 I.14.4 U=kspringx22 9 da 10 Yes Yes 10−2 I.15.3x x1=x−ut1−u2/c2 22 da, bf 10 No No 10−3 I.15.3t t1=t−ux/c21−u2/c2 20 da, bf 102 No No 10−4 I.15.10 p=m0v1−v2/c2 13 da, bf 10 No Yes 10−4 I.16.6 v1=u+v1+uv/c2 18 da, bf 10 No Yes 10−3 I.18.4 r=m1r1+m2r2m1+m2 17 da, bf 10 Yes Yes 10−2 I.18.12 τ = rF sin θ 15 da, bf 10 Yes Yes 10−3 I.18.16 L = mrv sin θ 17 da, bf 10 Yes Yes 10−3 I.24.6 E=14m(ω2+ω02)x2 22 da, bf 10 Yes Yes 10−4 I.25.13 Ve=qC 10 da 10 Yes Yes 10−2 I.26.2 θ1 = arcsin (n sin θ2) 530 da, bf-sin 102 Yes Yes 10−2 I.27.6 ff=11d1+nd2 14 da, bf 10 Yes Yes 10−2 I.29.4 k=ωc 8 da 10 Yes Yes 10−2 I.29.16 x=x12+x22−2x1x2cos(θ1−θ2) 2135 da, sym–, bf-squared 103 No No 10−4 I.30.3 I*=I*0sin2(nθ/2)sin2(θ/2) 118 da, bf 102 Yes Yes 10−3 I.30.5 θ=arcsin(λnd) 529 da, bf-sin 102 Yes Yes 10−3 I.32.5 P=q2a26πϵc3 13 da 10 Yes Yes 10−2 I.32.17 P=(12ϵcEf2)(8πr2/3)(ω4/(ω2−ω02)2) 698 da, bf-sqrt 10 No Yes 10−4 I.34.8 ω=qvBp 13 da 10 Yes Yes 10−2 I.34.10 ω=ω01−v/c 13 da, bf 10 No Yes 10−3 I.34.14 ω=1+v/c1−v2/c2ω0 14 da, bf 10 No Yes 10−3 I.34.27 E = ℏω 8 da 10 Yes Yes 10−2 I.37.4 I*=I1+I2+2I1I2cosδ 7032 da, bf 102 Yes No 10−3 I.38.12 r=4πϵℏ2mq2 13 da 10 Yes Yes 10−2 I.39.10 E=32pFV 8 da 10 Yes Yes 10−2 I.39.11 E=1γ−1pFV 13 da, bf 10 Yes Yes 10−3 I.39.22 PF=nkbTV 16 da, bf 10 Yes Yes 10−4 I.40.1 n=n0e−mgxkbT 20 da, bf 10 No Yes 10−2 I.41.16 Lrad=ℏω3π2c2(eℏωkbT−1) 22 da, bf 10 No No 10−5 I.43.16 v=μdriftqVed 14 da 10 Yes Yes 10−2 I.43.31 D = μekbT 11 da 10 Yes Yes 10−2 I.43.43 κ=1γ−1kbvA 16 da, bf 10 Yes Yes 10−3 I.44.4 E=nkbTln(V2V1) 18 da, bf 10 Yes Yes 10−3 I.47.23 c=γprρ 14 da, bf 10 Yes Yes 10−2 I.48.20 E=mc21−v2/c2 108 da, bf 102 No No 10−5 I.50.26 x = x1[ cos (ωt) + α cos (ωt)2] 29 da bf 10 Yes Yes 10−2
• Table 5 Tested Feynman equations, part 2 (same notation as in Table 4).

 Feynman Eq. Equation Solution Time(s) Methods Used Data Needed Solved ByEureqa Solved W/o da NoiseTolerance II.2.42 P=κ(T2−T1)Ad 54 da, bf 10 Yes Yes 10−3 II.3.24 FE=P4πr2 8 da 10 Yes Yes 10−2 II.4.23 Ve=q4πϵr 10 da 10 Yes Yes 10−2 II.6.11 Ve=14πϵpdcos θr2 18 da, bf 10 Yes Yes 10−3 II.6.15a Ef=34πϵpdzr5x2+y2 2801 da, sm, bf 104 No Yes 10−3 II.6.15b Ef=34πϵpdr3cos θsin θ 23 da, bf 10 Yes Yes 10−2 II.8.7 E=35q24πϵd 10 da 10 Yes Yes 10−2 II.8.31 Eden=ϵEf22 8 da 10 Yes Yes 10−2 II.10.9 Ef=σdenϵ11+χ 13 da, bf 10 Yes Yes 10−2 II.11.3 x=qEfm(ω02−ω2) 25 da, bf 10 Yes Yes 10−3 II.11.17 n=n0(1+pdEfcos θkbT) 28 da, bf 10 Yes Yes 10−2 II.11.20 P*=nρpd2Ef3kbT 18 da, bf 10 Yes Yes 10−3 II.11.27 P*=nα1−nα/3ϵEf 337 da bf-inverse 102 No Yes 10−3 II.11.28 θ=1+nα1−(nα/3) 1708 da, sym*, bf 102 No Yes 10−4 II.13.17 B=14πϵc22Ir 13 da 10 Yes Yes 10−2 II.13.23 ρc=ρc01−v2/c2 13 da, bf 102 No Yes 10−4 II.13.34 j=ρc0v1−v2/c2 14 da, bf 10 No Yes 10−4 II.15.4 E = − μMB cos θ 14 da, bf 10 Yes Yes 10−3 II.15.5 E = − pdEf cos θ 14 da, bf 10 Yes Yes 10−3 II.21.32 Ve=q4πϵr(1−v/c) 21 da, bf 10 Yes Yes 10−3 II.24.17 k=ω2c2−π2d2 62 da bf 10 No Yes 10−5 II.27.16 FE=ϵcEf2 13 da 10 Yes Yes 10−2 II.27.18 Eden=ϵEf2 9 da 10 Yes Yes 10−2 II.34.2a I=qv2πr 11 da 10 Yes Yes 10−2 II.34.2 μM=qvr2 11 da 10 Yes Yes 10−2 II.34.11 ω=g_qB2m 16 da, bf 10 Yes Yes 10−4 II.34.29a μM=qh4πm 12 da 10 Yes Yes 10−2 II.34.29b E=g_μMBJzℏ 18 da, bf 10 Yes Yes 10−4 II.35.18 n=n0exp(μmB/(kbT))+exp(−μmB/(kbT)) 30 da, bf 10 No Yes 10−2 II.35.21 M=nρμMtanh(μMBkbT) 1597 da, halve-input, bf 10 Yes No 10−4 II.36.38 f=μmBkbT+μmαMϵc2kbT 77 da bf 10 Yes Yes 10−2 II.37.1 E = μM(1 + χ)B 15 da, bf 10 Yes Yes 10−3 II.38.3 F=YAxd 47 da, bf 10 Yes Yes 10−3 II.38.14 μS=Y2(1+σ) 13 da, bf 10 Yes Yes 10−3 III.4.32 n=1eℏωkbT−1 20 da, bf 10 No Yes 10−3 III.4.33 E=ℏωeℏωkbT−1 19 da, bf 10 No Yes 10−3 III.7.38 ω=2μMBℏ 13 da 10 Yes Yes 10−2 III.8.54 pγ=sin(Etℏ)2 39 da, bf 10 No Yes 10−3 III.9.52 pγ=pdEftℏsin((ω−ω0)t/2)2((ω−ω0)t/2)2 3162 da, sym–, sm, bf 103 No Yes 10−3 III.10.19 E=μMBx2+By2+Bz2 410 da, bf-squared 102 Yes Yes 10−4 III.12.43 L = nℏ 11 da, bf 10 Yes Yes 10−3 III.13.18 v=2Ed2kℏ 16 da, bf 10 Yes Yes 10−4 III.14.14 I=I0(eqVekbT−1) 18 da, bf 10 No Yes 10−3 III.15.12 E = 2U(1 − cos (kd)) 14 da, bf 10 Yes Yes 10−4 III.15.14 m=ℏ22Ed2 10 da 10 Yes Yes 10−2 III.15.27 k=2παnd 14 da, bf 10 Yes Yes 10−3 III.17.37 f = β(1 + αcos θ) 27 bf 10 Yes Yes 10−3 III.19.51 E=−mq42(4πϵ)2ℏ21n2 18 da, bf 10 Yes Yes 10−5 III.21.20 j=−ρc0qAvecm 13 da 10 Yes Yes 10−2
• Table 6 Tested bonus equations.

Goldstein 8.56 is for the special case where the vectors p and A are parallel.

 Source Equation Solved Solved by Eureqa Methods used Rutherford scattering A=(Z1Z2αℏc4Esin2(θ2))2 Yes No da, bf-sqrt Friedman equation H=8πG3ρ−kfc2af2 Yes No da, bf-squared Compton scattering U=E1+Emc2(1−cos θ) Yes No da, bf Radiated gravitational wave power P=−325G4c5(m1m2)2(m1+m2)r5 No No – Relativistic aberration θ1=arccos(cos θ2−vc1−vccos θ2) Yes No da, bf-cos N-slit diffraction I=I0[sin(α/2)α/2sin(Nδ/2)sin(δ/2)]2 Yes No da, sm, bf Goldstein 3.16 v=2m(E−U−L22mr2) Yes No da, bf-squared Goldstein 3.55 k=mkGL2(1+1+2EL2mkG2cos(θ1−θ2)) Yes No da, sym–, bf Goldstein 3.64 (ellipse) r=d(1−α2)1+αcos(θ1−θ2) Yes No da, sym–, bf Goldstein 3.74 (Kepler) t=2πd3/2G(m1+m2) Yes No da, bf Goldstein 3.99 α=1+2ϵ2EL2m(Z1Z2q2)2 Yes No da, sym*, bf Goldstein 8.56 E=(p−qAvec)2c2+m2c4+qVe Yes No da, sep+, bf-squared Goldstein 12.80 E=12m[p2+m2ω2x2(1+αxy)] Yes Yes da, bf Jackson 2.11 F=q4πϵy2[4πϵVed−qdy3(y2−d2)2] No No – Jackson 3.45 Ve=q(r2+d2−2drcos α)12 Yes No da, bf-inv Jackson 4.60 Ve=Efcos θ(α−1α+2d3r2−r) Yes No da, sep*, bf Jackson 11.38 (Doppler) ω0=1−v2c21+vccos θω Yes No da, cos-input, bf Weinberg 15.2.1 ρ=38πG(c2kfaf2+H2) Yes Yes da, bf Weinberg 15.2.2 pf=−18πG[c4kfaf2+c2H2(1−2α)] Yes Yes da, bf Schwarz 13.132 (Klein-Nishina) A=πα2ℏ2m2c2(ω0ω)2[ω0ω+ωω0−sin2θ] Yes No da, sym/, sep*, sin-input, bf