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Mixing Differential Equations and Neural Networks for Physics-Informed Learning

eigenspace 7y ago
Very interesting stuff, I'll have to spend some time digesting this as it's relevant to some research interests of mine. This might be a dumb question, but one thing I'm having trouble understanding is your use of the phrase 'data driven' here. In the section "Solving Ordinary Differential Equations" you describe a method that seems to be able to (inefficiently) solve differential equations without having to take in some 'data', but instead only needing to know `f(u, t)`. In the later section, where you introduce the Physics Informed Neural Network, it seems (but I could be misunderstanding) that you are assuming I have measurements of the actual solution and we're training the network with that data. Is this correct? I was initially imagining a method where say I know the solution to the linear part of the differential equation and then train the network just using the non-linear part without having to have any data.
- ChrisRackauckas 7y ago
  >In the later section, where you introduce the Physics Informed Neural Network, it seems (but I could be misunderstanding) that you are assuming I have measurements of the actual solution and we're training the network with that data. Is this correct? Yes, this is the PINN method M.Raissi, P.Perdikaris, and G.E.Karniadakis: use a physical underpinning as a function to help the learning but relax towards the data. >I was initially imagining a method where say I know the solution to the linear part of the differential equation and then train the network just using the non-linear part without having to have any data. These are the mixed neural differential equations that I have been showcasing with DiffEqFlux.jl https://github.com/JuliaDiffEq/DiffEqFlux.jl#mixed-neural-des https://github.com/JuliaDiffEq/DiffEqFlux.jl#mixed-neural-de... So it's just different methods by different people / different groups (for different purposes).

Very interesting stuff, I'll have to spend some time digesting this as it's relevant to some research interests of mine. This might be a dumb question, but one thing I'm having trouble understanding is your use of the phrase 'data driven' here. In the section "Solving Ordinary Differential Equations" you describe a method that seems to be able to (inefficiently) solve differential equations without having to take in some 'data', but instead only needing to know `f(u, t)`. In the later section, where you introduce the Physics Informed Neural Network, it seems (but I could be misunderstanding) that you are assuming I have measurements of the actual solution and we're training the network with that data. Is this correct? I was initially imagining a method where say I know the solution to the linear part of the differential equation and then train the network just using the non-linear part without having to have any data.

ChrisRackauckas 7y ago
>In the later section, where you introduce the Physics Informed Neural Network, it seems (but I could be misunderstanding) that you are assuming I have measurements of the actual solution and we're training the network with that data. Is this correct? Yes, this is the PINN method M.Raissi, P.Perdikaris, and G.E.Karniadakis: use a physical underpinning as a function to help the learning but relax towards the data. >I was initially imagining a method where say I know the solution to the linear part of the differential equation and then train the network just using the non-linear part without having to have any data. These are the mixed neural differential equations that I have been showcasing with DiffEqFlux.jl https://github.com/JuliaDiffEq/DiffEqFlux.jl#mixed-neural-des https://github.com/JuliaDiffEq/DiffEqFlux.jl#mixed-neural-de... So it's just different methods by different people / different groups (for different purposes).