[HN Gopher] Finding paths of least action with gradient descent ...
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       Finding paths of least action with gradient descent (2023)
        
       Author : E-Reverance
       Score  : 27 points
       Date   : 2025-04-26 06:24 UTC (3 days ago)
        
 (HTM) web link (greydanus.github.io)
 (TXT) w3m dump (greydanus.github.io)
        
       | constantcrying wrote:
       | >Some, like the double pendulum or the three-body problem, are
       | deterministic but chaotic. In other words, their dynamics are
       | predictable but we can't know their state at some time in the
       | future without simulating all the intervening states.
       | 
       | Literal nonsense. Everything in the second sentence is false.
       | 
       | Deterministic means that the state at some point in time fixes
       | the state at all future points in time. Nevertheless in a
       | deterministic system you can know a future state without
       | calculating intermediary states.
       | 
       | Chaotic means that future states are discontinuous in regards to
       | the initial state. Nevertheless a chaotic system can be known at
       | future states without calculating intermediary states, you can
       | even have an _analytic_ solution to a chaotic system. Furthermore
       | chaotic can mean that you _can 't_ calculate future states from
       | initial states. Numerical ODE solvers in particularly have errors
       | which grow exponential in time. So simulating intermediate states
       | does not give you the solution to the problem.
        
       | imtringued wrote:
       | While the idea is obviously correct, the paper itself suffers
       | from extremely sloppy writing.
       | 
       | They discretize the integral with a discrete sum, but then forget
       | to discretize the variables by substituting x with x(t_i) or at
       | least x_i, same for dot x. They put the objective function x hat
       | = argmin S(X) last, when it is the most important aspect.
       | 
       | In the equation where x hat must fulfill the Euler lagrange
       | equation for all t, they butchered the application of the
       | derivative with respect to a constant point.
       | 
       | It should look more like this:
       | 
       | https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe...
       | 
       | You need to explicitly pass in the x(t), dot x (t) and t as
       | arguments into the derivative. Their notation implies that you
       | have to take the derivative with respect to a constant (not at a
       | point) which always returns zero (a blatantly banal property) or
       | that the function (=the laws of physics) behind x(t) varies over
       | time (shudder).
       | 
       | Overall this was extremely unpleasant to read even though the
       | approach is neat.
        
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       (page generated 2025-04-29 23:00 UTC)