Machine Learning for Physical Systems · Lecture 6

Fields and rubber

Shortcut: P

Physics in the loss or built in: a heated rod, a mass that swings for ten periods, and a rubber model.

    The temperature of the rod over position and time

    The temperature along the rod at one time
    The rod in scaled units: ut = uxx, start u = sin πx, both ends held at u = 0. Exact: u = e−π²t sin πx. For a 10 cm aluminium rod, t = 0.2 is 21 s.
    1. Compare 21 and 101 points: error and time.
    2. Choose SI units: what does the network find?
    3. Show the error: where is each one least accurate?

    Crank-Nicolson: points along the rod

    Network: start and boundary values

    Crank-Nicolson, in this page
    Network, training

    Loss terms of the network over its training (log scale)

    The terms of the loss over the training steps

    More options
    Crank-Nicolson, timed again

    Every solve includes its setup. The time shown is the average over repeated solves.

    Check the rubber fit

    Checks that no small change of the weights, within the constraint, lowers the squared error.

    Table: the rod, times and errors
    MethodTimeLargest errorRelative error
    Table: the rubber data (1944) and the fit
    The fit
    TermWeight
    How it is computed

    Rubber data: L. R. G. Treloar (1944), Trans. Faraday Soc. 40, 59–70, nominal stress against stretch of one vulcanised rubber, as digitised by P. Steinmann, M. Hossain and G. Possart (2012). The networks of steps 1 and 2 were trained for the lecture (PyTorch, 64 bit, CPU); their fields, curves and loss terms are stored in this page.