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Input:Target evolutionary PDE equation 2; initial network uθ; end time T Output :The predicted model uθn (x) at each timestamp tn 1 Set hyper-parameters: timestamps number Nt, number of maximum training iterations
2 Step (a): learn ugo(x) by PINN ;Mo, M,learning rate n, threshold value ∈ ;
3 fori=1,2,..,Mo do
4Compute the mean square error loss Lo(0°); Update the parameter 00 via gradient descent 0+1 = 0 - nVL°(0) ;
5
Step(b): denote 0* = 0Mo and learn uo1(x),., uon (x),.. sequentially by transfer learning ; for n =0,1,2,.,Nt -1 do
8for i=1,2,..,M do
9Compute lossCn+1(0n+1) by equation 5;
1mUpdate the parameter 0n+1 via gradient descent 0n+1 0+1-nVLn+1（0%+1)； 计+1
if|cn+1(07+1）-Cn+1(07+1)/<cthen
12L denote θ𝑛+1 = 0n+1 and break;
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MethodL2RDEqtimeL² ACEqtimeL2KS EqimeL2 NS Eq time
Original PINN4.17e-0213978.23e-0114121.00e+0011.32e+001
Adaptive sampling1.65e-0215618.64e-0314609.98e-0169018.45e-0125385
Self-attention1.14e-0214501.05e-0117708.22e-0154159.28e-0121296
Time marching3.98e-0332152.01e-0237158.02e-0155278.85e-0126200
Causal PINN3.99e-0573581.66e-0392644.16e-02220294.73e-02 5 days
TL-DPINN1 (ours)1.82e-0514635.92e-0423287.17e-0350503.44e-0212440
TL-DPINN2 (ours)9.34e-057489.82e-0411003.55e-0251713.66e-0256875
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MethodL2 rrRD Eq.L² rrAC Eq.
timetime
Causal PINN3.73e-05 ± 4.66e-067207± 2191.51e-03 ± 2.12e-049060± 341
TL-DPINN11.76e-05 ± 1.06e-061463 ± 536.08e-04 ± 3.06e-052328±89
TL-DPINN29.89e-05 ± 8.94e-06811 ± 1229.29e-04 ± 8.06e-051291 ± 178
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Methodrn
Reaction-DiffusionNavier-Stokes
Casual PINN61.7052.3326.242.77
TL-DPINN1439.37384.47259.208.32
TL-DPINN2276.40239.52127.556.37
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MethodL2RDEqimeL2 AC Eqtime
Forward Euler1.32e-032089.57e-03304
Backward Euler2.74e-032061.64e-02444
2nd RK1.97e-037611.17e-031054
4th RK2.11e-0311871.31e-031779
TL-DPINN11.82e-0514635.92e-042328
TL-DPINN29.34e-057489.82e-041100
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MethodL2RDEqtimeL AC Eqime
Without TL4.01e-0458801.35e-029170
last layer3.31e-046381.01e-023624
last 2 layers3.22e-042211.01e-024029
last 3 layers4.08e-042321.01e-024685
TL-DPINN11.82e-0514635.92e-042328
TL-DPINN29.34e-057489.82e-041100
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NotationMeaning
PINNPhysics-informed neural network
PDEPartial differential equation
TL-DPINNTransfer learning enhanced discrete PINN
TL-DPINN1Crank-Nicolson time differencing in TL-DPINN
TL-DPINN1Gauss-Legendre time differencing in TL-DPINN
Lor LnPhysics-informed loss function
Differential operator, such as N(u) = Ucr
RThe residual term of the evolutionary PDE, for example R(u) = ut - Uxx
ΩSpatial domain
8The boundary of the spatial domain
TEnd time
NtTimestamps number
NbThe collocation points number on ∂Ω
Nu,NrThe collocation points number in Ω or Ω × [0,T]
u(t,x）The exact solution to the evolutionary PDE
un(x)The time differencing scheme solution to the evolutionary PDE
ugn(x)The discrete PINN solution to the evolutionary PDE
hjThe j component in the output of the last hidden layer of the neural network
x,xr,xb tortnSpatial coordinate
0 or 0n,Wn,wnTemporal coordinate Neural network parameters
△t or TTime step, the interval time between two adjacent timestamps
Mo,M1
Number of maximum iterations in different training stages
nThe learning rate in gradient descent methods
EThe threshold value
I- The L² norm of a function, defined by |lfll = (Jo If(x)|²dx)²
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