| 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 |
| 4 | Compute 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 |
| 8 | for i=1,2,..,M do | |
| 9 | Compute lossCn+1(0n+1) by equation 5; | |
| 1m | | Update 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 |
| 12 | L denote θ𝑛+1 = 0n+1 and break; | |
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"html": "| Notation | Meaning |
| PINN | Physics-informed neural network |
| PDE | Partial differential equation |
| TL-DPINN | Transfer learning enhanced discrete PINN |
| TL-DPINN1 | Crank-Nicolson time differencing in TL-DPINN |
| TL-DPINN1 | Gauss-Legendre time differencing in TL-DPINN |
| Lor Ln | Physics-informed loss function |
| Differential operator, such as N(u) = Ucr |
| R | The residual term of the evolutionary PDE, for example R(u) = ut - Uxx |
| Ω | Spatial domain |
| 8 | The boundary of the spatial domain |
| T | End time |
| Nt | Timestamps number |
| Nb | The collocation points number on ∂Ω |
| Nu,Nr | The 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 |
| hj | The j component in the output of the last hidden layer of the neural network |
| x,xr,xb tortn | Spatial coordinate |
| 0 or 0n,Wn,wn | Temporal coordinate Neural network parameters |
| △t or T | Time step, the interval time between two adjacent timestamps |
| Mo,M1 | |
| Number of maximum iterations in different training stages |
| n | The learning rate in gradient descent methods |
| E | The threshold value |
| I- | The L² norm of a function, defined by |lfll = (Jo If(x)|²dx)² |
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