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Algorithm1:Numerical Integration
Input initial state x,start timeto,end time T,error tolerance etol,initial
stepsize h. Initialize z(O) = x,t = to
Whilet<T error_est=
While error_est>etol h←h×DecayFactor
,error_est=yh(t,z) If error_est<etol h←h×IncreaseFactor
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NaiveAdjointACAMALI
ComputationNzNf×Nt×m×2NzNf ×(Nt+Nr)×mNzNf×Nt×(m+1)NzNf×Nt×(m+2)
MemoryNzNf ×Nt × mNzNfNz(Nf+Nt)Nz(Nf +1)
Computation graph depthNf×Nt × mNf × NrNf×NtNf×Nt
Reverse accuracyX
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Input (zin, Uin, Sin,h) where Sin is current time, Zin and Uin are correponding values at time Sin,h is stepsize.
Forward S1 = Sin +h/2
k1 = Zin + Uin × h/2
u1 = f(k1,S1)
Uout = Uin + 2(u1 - Uin)
Zout = k1 + Uout × h/2
Sout = S1 +h/2 Output (Zout,Vout, Sout,h)
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Input (zout, Uout, Sout,h) where Sout is current time, Zout and vout are corresponding values at Sout,h is stepsize. Inverse S1 = Sout -h/2 k1 = Zout - Uout X h/2 u1 = f(k1,S1)
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Algorithm 4: MALI to acheive accuracy at a constant memory cost w.r.t integration time
Input Initial state zo, start time to, end time T
Forward Apply the numerical integration in Algo.1, with the function defined by Algo. 2.
Delete computation graph on the fly, only keep end-time state (zNt, UNt）
Keep accepted discretized time points {ti}N=0 (ignore processto search for optimal stepsize) Backward
8L by Eq.3,initialize dL =0
Initialize a(T) = (T) d
For i in {Nt,Nt -1,..,2,1}:
Reconstruct (zi-1, Ui-1) from (zi,Ui) by Algo.7 Local forward (zi,Ui,ti,hi)= γ(zi-1,Ui-1,ti-1,hi)
Local backward, get Of(zi-1,ti-1,0) and f（2i-1ti-1,0)
dzi-1 80 dL
Update a(t) and byEq.2 and Eq. 3 discretized at time points ti-1 and ti de
Delete local computation graph
Output the adjoint state a(to) (gradient w.r.t input zo) and parameter gradient
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∈=1/255∈=2/255
MALIHeun-EulerRK23Dopri5MALIHeun-EulerRK23Dopri5
Neural ODEMALI14.6914.7214.7715.7110.3810.4610.6210.62
Heun-Euler14.7714.7514.8015.7410.6310.4710.4410.49
RK2314.8214.7714.7915.6910.7810.5310.4810.56
Dopri514.8214.7814.7915.1510.7610.4910.4810.51
ResNet13.029.57
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Fixed-stepsize solvers of various stepsizesAdaptive-stepsize solver of various tolerances
Stepsize10.50.250.150.1Tolerance1.00E+001.00E-011.00E-02
Neural ODEMALI42.3366.469.5970.1769.94MALI62.5669.8969.87
Euler21.9461.2567.3868.6970.02Heun-Euler68.4869.8769.88
RK242.336969.7270.1469.92RK2350.7769.8969.93
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ResNet
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MethodAccuracy (%)
Adjoint392.8± 0.4
SemiNorm392.9 ±0.4
Naive ACA93.2±0.2
MALI93.2±0.2
93.7 ±0.3
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Percentage of training dataRNN1RNN-GRU1Latent-ODE
Adjoint1Naive2ACA²MALI
10%2.4511.9720.4710.3620.3120.35
20%1.7111.4210.4410.30²0.2720.27
50%0.7910.7510.4010.2920.2620.26
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DatasetContinuous Flow (FFJORD)Discrete Flow
Vanilla4RNODE5SemiNorm3MALIRealNVP6i-ResNetGlow8Flow++9Residual Flow10
MNIST0.9940.9750.9630.871.0661.0571.058-0.9710
CIFAR103.4043.3853.3533.273.4963.4573.3583.2893.2810
ImageNet64-3.835-3.713.986-3.818=3.7610
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Test Accuracy on Speech Commands (Higher is better)93.7 ± 0.393.7 ± 0.193.5± 0.293.7 ± 0.3
Test MSEof latent ODE on Mujoco (Lower is better)10% training data0.350.360.330.33
20% training data0.270.250.260.27
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