Learning in time varying, dissipative electrical networks

1. A method, comprising:

determining a trajectory for a dissipative learning network, the dissipative learning network including a plurality of neuron banks and at least one weighting array, the at least one weighting array being between a first neuron bank and a second neuron bank of the plurality of neuron banks, at least one of the plurality of neuron banks or the at least one weighting array including an electrical component that dissipates energy, the trajectory including a plurality of states of the dissipative learning network at a first plurality of times and being for at least one input signal;

determining a perturbed trajectory for the dissipative learning network based on a plurality of target outputs, the perturbed trajectory including a plurality of perturbed states of the dissipative learning network at a second plurality of times, being based on the plurality of target outputs, and being for the at least one input signal;

determining gradients for a portion of the dissipative learning network based on the trajectory and the perturbed trajectory, the portion of the dissipative learning network including the at least one weighting array, the determining the gradients further includes

fractionally integrating the trajectory to provide a fractionally integrated trajectory;

fractionally integrating the perturbed trajectory to provide a fractionally integrated perturbed trajectory; and

determining a difference between the fractionally integrated trajectory and the fractionally integrated perturbed trajectory for each weight in the plurality of weighting arrays, the difference corresponding to a gradient for each weight; and

adjusting the portion of the dissipative learning network based on the gradients.

2. The method of claim 1 , wherein the determining the trajectory further includes:

sampling the trajectory at a first time and a second time to identify a first sampled trajectory and a second sampled trajectory;

wherein the determining the perturbed trajectory further includes sampling the perturbed trajectory at a third time and a fourth time to identify a first sampled perturbed trajectory and a second sampled perturbed trajectory;

wherein the fractionally integrating the trajectory includes interpolating between the first sampled trajectory and the second sampled trajectory to provide a fractionally integrated sampled trajectory;

wherein the fractionally integrating the perturbed trajectory includes interpolating between the first sampled perturbed trajectory and the second sampled perturbed trajectory to provide a fractionally integrated sampled perturbed trajectory; and

wherein the determining the difference further includes determining a first difference between the fractionally integrated sampled trajectory and the fractionally integrated sampled perturbed trajectory for each weight in the at least one weighting array, the first difference corresponding to a gradient for each weight.

3. The method of claim 1 , wherein the determining the trajectory includes:

providing the at least one input signal to the dissipative learning network; and

sampling the at least one weighting array while the at least one input signal is provided.

4. The method of claim 1 , wherein each of the plurality of neuron banks includes a plurality of neurons including plurality of circuit elements, the plurality of circuit elements including at least one nonlinear circuit element, at least one dissipative circuit element, and at least one time varying circuit element.

5. The method of claim 4 , wherein the plurality of circuit elements are selected from capacitive circuit elements, inductive circuit elements, fractional memristive circuit elements and resistive circuit elements.

6. A method, comprising

determining a trajectory for a dissipative learning network, the dissipative learning network including a plurality of neuron banks and at least one weighting array, the at least one weighting array being between a first neuron bank and a second neuron bank of the plurality of neuron banks, at least one of the plurality of neuron banks or the at least one weighting array including an electrical component that dissipates energy, the trajectory including a plurality of states of the dissipative learning network at a first plurality of times and being for at least one input signal, wherein determining the trajectory further includes

providing the at least one input signal to the dissipative learning network; and

sampling the plurality of weighting arrays while the at least one input signal is provided;

determining a perturbed trajectory for the dissipative learning network based on a plurality of target outputs, wherein the determining the perturbed trajectory further includes:

coupling the dissipative learning network to a loss circuit, wherein the loss circuit determines a difference between at least one output signal from the dissipative learning network and the target outputs and provides clamping signals to the dissipative learning network based on the difference;

providing the at least one input signal to the dissipative learning network while the dissipative learning network is coupled to the loss circuit; and

sampling the at least one weighting array while the at least one input signal is provided to the dissipative learning network and the dissipative learning network is coupled to the loss circuit;

determining gradients for a portion of the dissipative learning network based on the trajectory and the perturbed trajectory; and

adjusting the portion of the dissipative learning network based on the gradients.

7. A system, comprising:

a dissipative learning network characterized by an action and having a trajectory corresponding to at least one input signal, the dissipative learning network including a plurality of neuron banks, at least one weighting array, and at least one output, the at least one weighting array being between a first neuron bank and a second neuron bank of the plurality of neuron banks, at least one of the plurality of neuron banks or the at least one weighting array including an electrical component that dissipates energy, the trajectory including a plurality of states of the dissipative learning network at a first plurality of times and being for the at least one input signal;

loss circuitry selectively connectable to the dissipative learning network, the loss circuitry providing a perturbed trajectory for the dissipative learning network based on a plurality of target outputs and the at least one input signal, the perturbed trajectory including a plurality of perturbed states of the dissipative learning network for a second plurality of times, being based on the plurality of target outputs, and being for the at least one input signal, the loss circuitry monitoring the at least one output of the dissipative learning network and injecting at least one perturbation to the at least one output; and

a gradient estimator coupled with the dissipative learning network, wherein the gradient estimator determines gradients for a portion of the dissipative learning network based on the trajectory and the perturbed trajectory and adjusts the portion of the dissipative learning network based on the gradients, wherein the gradient estimator further

fractionally integrates the trajectory to provide a fractionally integrated trajectory;

fractionally integrates the perturbed trajectory to provide a fractionally integrated perturbed trajectory; and

determines a difference between the fractionally integrated trajectory and the fractionally integrated perturbed trajectory for each weight in the plurality of weighting arrays, the difference corresponding to a gradient for each weight.

8. The system of claim 7 , wherein to determine the trajectory, the gradient estimator samples the trajectory at a first time and a second time to identify a first sampled trajectory and a second sampled trajectory;

wherein to determine the perturbed trajectory the gradient estimator further samples the perturbed trajectory at a third time and a fourth time to identify a first sampled perturbed trajectory and a second sampled perturbed trajectory;

wherein to fractionally integrate the trajectory, the gradient estimator interpolates between the first sampled trajectory and the second sampled trajectory to provide a fractionally integrated sampled trajectory;

wherein to fractionally integrate the perturbed trajectory, the gradient estimator interpolates between the first sampled perturbed trajectory and the second sampled perturbed trajectory to provide a fractionally integrated sampled perturbed trajectory; and

wherein to determine the difference, the gradient estimator further determines a first difference between the fractionally integrated sampled trajectory and the fractionally integrated sampled perturbed trajectory for each weight in the at least one weighting array, the first difference corresponding to a gradient for each weight.

9. The system of claim 7 , wherein the gradient estimator further samples the at least one weighting array to determine the gradients based on the perturbed trajectory and the trajectory, the gradient estimator configured to adjust the at least one weighting array based on the gradients.

10. The system of claim 7 , wherein each of the plurality of neuron banks includes a plurality of neurons including plurality of circuit elements, the plurality of circuit elements including at least one nonlinear circuit element, at least one dissipative circuit element, and at least one time varying circuit element.

11. The system of claim 7 , wherein each of the at least one weighting array includes programmable resistances.

12. A system, comprising:

a dissipative learning network characterized by an action and having a trajectory corresponding to at least one input signal, the dissipative learning network including a plurality of neuron banks and at least one weighting array, the at least one weighting array being between a first neuron bank and a second neuron bank of the plurality of neuron banks, at least one of the plurality of neuron banks or the at least one weighting array including an electrical component that dissipates energy, the plurality of neuron banks including a plurality of neurons including plurality of circuit elements, the plurality of circuit elements including at least one nonlinear circuit element, at least one dissipative circuit element, and at least one time varying circuit element, wherein the plurality of circuit elements are selected from capacitive circuit elements, inductive circuit elements, fractional memristive circuit elements and resistive circuit elements, the trajectory including a plurality of states of the dissipative learning network at a first plurality of times and being for the at least one input signal;

loss circuitry selectively connectable to the dissipative learning network, the loss circuitry providing a perturbed trajectory for the dissipative learning network based on a plurality of target outputs and the at least one input signal, the perturbed trajectory including a plurality of perturbed states of the dissipative learning network for a second plurality of times, being based on the plurality of target outputs, and being for the at least one input signal; and

a gradient estimator coupled with the dissipative learning network, wherein the gradient estimator determines gradients for a portion of the dissipative learning network based on the trajectory and the perturbed trajectory and adjusts the portion of the dissipative learning network based on the gradients.

13. A method, comprising:

determining a trajectory for a dissipative learning network, the trajectory being in response to at least one input signal, the dissipative learning network being characterized by an action and including a plurality of neuron banks and at least one weighting array, the at least one weighting array being between a first neuron bank and a second neuron bank of the plurality of neuron banks, at least one of the plurality of neuron banks or the at least one weighting array including an electrical component that dissipates energy, the trajectory including a plurality of states of the dissipative learning network at a first plurality of times and for the at least one input signal;

determining a perturbed trajectory for the dissipative learning network based upon a loss function for a plurality of target outputs and the at least one input signal, the perturbed trajectory including a plurality of perturbed states of the dissipative learning network at a second plurality of times, being based on the loss function, and being for the at least one input signal; and

adjusting a portion of the dissipative learning network based on the trajectory and the perturbed trajectory such that a minimum of the action corresponds to a loss function minimum, the adjusting the loss function further including

determining gradients for the portion of the dissipative learning network based on the trajectory and the perturbed trajectory, the gradients corresponding to a convergence between the minimum of the action and the loss function minimum, determining the gradients further including

fractionally integrating the trajectory to provide a fractionally integrated trajectory;

fractionally integrating the perturbed trajectory to provide a fractionally integrated perturbed trajectory; and

determining a difference between the fractionally integrated trajectory and the fractionally integrated perturbed trajectory; and

adjusting the portion of the dissipative learning network based on the gradients.

14. A method, comprising:

determining a trajectory for a dissipative learning network, the trajectory being in response to at least one input signal, the dissipative learning network being characterized by an action and including a plurality of neuron banks and at least one weighting array, the at least one weighting array being between a first neuron bank and a second neuron bank of the plurality of neuron banks, at least one of the plurality of neuron banks or the at least one weighting array including an electrical component that dissipates energy, the trajectory including a plurality of states of the dissipative learning network at a first plurality of times and for the at least one input signal, wherein the dissipative learning network includes a plurality of circuit elements, the plurality of circuit elements including at least one nonlinear circuit element, at least one dissipative circuit element, and at least one time varying circuit element;

determining a perturbed trajectory for the dissipative learning network based upon a loss function for a plurality of target outputs and the at least one input signal, the perturbed trajectory including a plurality of perturbed states of the dissipative learning network at a second plurality of times, being based on the loss function, and being for the at least one input signal; and

adjusting a portion of the dissipative learning network based on the trajectory and the perturbed trajectory such that a minimum of the action corresponds to a loss function minimum, wherein the adjusting further includes

determining gradients for the portion of the dissipative learning network based on the trajectory and the perturbed trajectory, the gradients corresponding to a convergence between the minimum of the action and the loss function minimum; and

adjusting the portion of the dissipative learning network based on the gradients.