Trajectory optimization is cool. The idea is to take a dynamical problem as a big ole optimization problem, finding the best actions to take to achieve your goals or maximize a reward.

There are a couple of flavors of trajectory optimization (shooting methods, collocation methods) http://www.matthewpeterkelly.com/tutorials/trajectoryOptimization/

PyTorch gives a pretty low overhead extension to Numpy that also gives autodifferentiation. It is mainly intended as a neural network library, for which it has a number of facilities.

Gradient Descent is not the preferred method for these problems (According to Boyd’s Convex optimization course). Gradient Descent has shit convergence compared to newton iteration, but is very flexible and easy to implement.

In addition, using a normal ODE solver from Scipy would be much more stable, but it would require cleverness to have the same code work for both scipy and the torch parts. So screw it.

One nicety of this approach is that we don’t even have to have our derivatives solved for. They could be all tied up in a

I thought that maybe I could just weight the dynamics cost enough to have it require the dynamics be satisfied, but that did not seem to work. Maybe with more fiddling? On further review my code had massive bugs in it. I’m not sure that the dynamics cost version wouldn’t work, but the Lagrange multiplier method seems to work well and makes sense too.

In this formulation, we can also train some kind of parametrized controller function $f_w(x)$ by sampling some random initial starting conditions (or even dynamical parameters like mass and length etc, or noise forces). This is quite nice.

Additional bits that may be nice: Backtracking line search, logarithmic potential for inequalities, I wonder if a symplectic style interleaving of position and momentum might be nice even for this global case. Should definitely just tie up all the vars into a single x. Can we use a lagrangian or hamiltonian and then have pytorch differentiate that? It may in fact be nice to use some combinator to be able to hand the same function to ODEInt for a couple reasons (getting good initilizations  of the path for example).

For a simple system, I’m using $\dot{x}=v$ , $\dot{v}=f$ , where you get to control f at every time point and x is starting at 0 and wants to get to 1. I’m using a simple scheme of finite difference in time for the time derivative. x and v are defined at $t$ and $f, lx, lv$ are defined at the half time steps $t + \frac{1}{2}$. You need at least two time steps to get a derivative. I’m adding a square cost to the force, otherwise it would just get a huge force. lx and lv are Lagrange multipliers enforcing the equations of motion at each time step

Here was an initial pass (just here for for historical reasons, look at the updated one below. This one does not work as is)

import torch

batch = 1
N = 10
T = 5
dt = T/N

#print(x)

def calc_loss(x,v,f):

delx = (x[:,1:] - x[:, :-1]) / dt
delv = (v[:,1:] - v[:, :-1]) / dt

xbar = (x[:,1:] + x[:, :-1]) / 2
vbar = (v[:,1:] + v[:, :-1]) / 2

dxdt = vbar
dvdt = f

xres = xbar - dxdt
vres = vbar - dvdt

dyn_err = torch.sum(torch.abs(xres) + torch.abs(vres), dim=1) #torch.sum(xres**2 + vres**2, dim=1) # + Abs of same thing?

reward = torch.sum((xbar-1)**2 + f**2, dim=1)

total_cost = 100 * dyn_err + reward

import torch.optim as optim
'''
optimizer = optim.SGD(net.parameters(), lr=0.01)

output = net(input)
loss = criterion(output, target)
loss.backward()
optimizer.step()    # Does the update
'''

#Could interleave an ODE solve step - stinks that I have to write dyanmics twice
#Or interleave a sepearate dynamic solving
# Or could use an adaptive line search. Backtracking
# Goal is to get dyn_err quite small

learning_rate = 0.001
for i in range(40):
total_cost=calc_loss(x,v,f)
total_cost.backward()
while dyn_loss > 0.01:
dyn_loss.backward()
x[:,1:] -= learning_rate * x.grad[:,1:] # Do not change Starting conditions
reward.backward()

print(x)
print(v)
print(f)


goofed up a couple things (inlcuding my xres making no sense. You need to explicility zero gradients. Pretty annoying). Lagrange multiplier method makes total sense.

Could we use a Hamiltonian and use autograd to derive equations of motion? Seems plausible and convenient.

Can I make a custom pytorch layer for sparse Hessians? The data oriented viewpoint would have you pump the gradient and hessian backward. Or could you automatically build an H-matrix structure for the hessian of convnets?

Put a neural controller in there. Each batch could have randomized parameters, noise, and initial conditions.

Is rebuilding total_cost every time bad?

import torch

batch = 1
N = 20
T = 5
dt = T/N

#print(x)

'''
class Vars():
def __init__(self, N=10):
self.data = torch.zeros(batch, N, 2)
self.data1 = torch.zeros(batch, N-1, 3)
self.lx = self.data1[:,:,0]
self.lv = self.data1[:,:,1]
self.f = self.data1[:,:,2]
self.x = self.data[:,:,0]
self.v = self.data[:,:,1]
'''
def calc_loss(x,v,f, lx, lv):
#finite difference derivative
delx = (x[:,1:] - x[:, :-1]) / dt
delv = (v[:,1:] - v[:, :-1]) / dt
#average at half time step
xbar = (x[:,1:] + x[:, :-1]) / 2
vbar = (v[:,1:] + v[:, :-1]) / 2

dxdt = vbar
dvdt = f - xbar

#residual of dynamics (want these numbers to be zeor)
xres = delx - dxdt
vres = delv - dvdt

#dyn_err = torch.sum(torch.abs(xres) + torch.abs(vres), dim=1) #torch.sum(xres**2 + vres**2, dim=1) # + Abs of same thing?

lagrange_mult = torch.sum(lx * xres + lv * vres, dim=1)

cost = torch.sum( torch.abs(x-1), dim=1) + torch.sum( f**2, dim=1) #+
#cost = torch.sum((x-1)**2, dim=1)

total_cost =   cost + lagrange_mult  #100 * dyn_err + reward

import torch.optim as optim
'''
optimizer = optim.SGD(net.parameters(), lr=0.01)

output = net(input)
loss = criterion(output, target)
loss.backward()
optimizer.step()    # Does the update
'''

#Could interleave an ODE solve step - stinks that I have to write dyanmics twice
#Or interleave a sepearate dynamic solving
# Or could use an adaptive line search. Backtracking
# Goal is to get dyn_err quite small

'''
learning_rate = 0.001
for i in range(40):
total_cost=calc_loss(x,v,f)
total_cost.backward()
while dyn_loss > 0.01:
dyn_loss.backward()
x[:,1:] -= learning_rate * x.grad[:,1:] # Do not change Starting conditions
reward.backward()
'''
learning_rate = 0.001
for i in range(10000):
total_cost = calc_loss(x,v,f, lx, lv)
print(total_cost)
#print(x)

total_cost.backward()