Introduction for Quantum Oscillator problem#

The goal of this project was not so much to implement the network itself, but to verify how efficiently differential equations with an eigenproblem can be solved using neural networks. Unfortunately, as a result of this experiment, we came to the conclusion that the process of learning network is too time-consuming and in this form can not be applied to other problems that we considered as a target.

As sad as it sounds, we leave the implementation of the network for all interested parties to see. On the other hand, we ourselves are changing the space of our interest from equations with an eigenproblem to equations without one.

Maths behind the problem#

Consider a mathematical problem of the following form

$$ \mathcal{L}f(x) = \lambda f(x) $$

(1)

$x$ a variable
$\mathcal{L}$ is a differential operator which depends on $x$ and its derivatives
$f(x)$ is the eigenfunction
$\lambda$ is the eigenvalue associated with eigenfunction

Network input-output#

Our neural network in form of represented by $N(x, \lambda )$ is expected to get two values as input:

$x$ - single floating point value within $x_L$ to $x_R$ range
$\lambda$ - eigenvalue approximation selected by neural network

As output, we also expect two values:

$y$ - floating point value corresponding to single $x$ value within $x_L$ to $x_R$ range - $f(x)$ approximation
$\lambda$ - eigenvalue approximation selected by neural network

To use our neural network as a replacement for $f(x)$ we need to put the values returned from it into the equation below

$$ f(x,λ) = f_b + g(x)N(x,λ) $$

(2)

$f_b$ arbitrary constant
$g(x)$ boundary condition
$N(x, \lambda )$ - our neural network

$$ g(x) = (1 −e^{−(x−x_L)})(1 − e^{−(x−x_R)}) $$

(3)

$x_L$ minimum (left) value in the examined range
$x_R$ maximum (right) value in the examined range

Loss function#

$$ \mathcal{L}f(x) - λf(x) = 0 $$

(4)

By rearranging (1) to form (4) we are able to use it to measure how far from exact eigenfunction and exact eigenvalue is current state of neural network. To achieve this we have to replace $f(x)$ with $f(x, \lambda )$ and use $\lambda$ retuned from network.

Therefore without any prepared input data, just based on loss function and back propagation we can move from a random state of network to an approximation of the equation solution.

Regularizators#

To find multiple solutions of given equation with this neural network, it is necessary to add additional components to the loss function which will prevent the neural network from staying in trivial solution state and force transitions between successive states.

$$ L_f = \frac{1}{f(x, \lambda )^2} $$

(5) - avoid trivial solutions

$$ L_{\lambda} = \frac{1}{\lambda^2} $$

(6) - avoid trivial solutions

$$ L_{drive} = e ^ {-\lambda + c} $$

(7) - push to subsequent solutions

$c$ - variable increased in regular intervals

Problem we are solving#

To make use of above theory we created network for solving Schrodinger's equation. In our case it takes following form

$$ \left[ \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) \right] \psi(x) = E\psi(x) $$

(8)

With $V(x)$ defined as

$$ \frac{1}{2}kx^2 $$

(9)

The exact solution of this equation has following form

$$ \psi _n(x) = \frac{1}{\sqrt{2^nn!}}\frac{e^{-\frac{x^2}{2}}}{\pi ^ {\frac{1}{4}}}H_n(x) \;\;\;\;\;\;\;\;\;\;\; E_n = n + \frac{1}{2} $$

(10)

Network structure#

The network is a simple sequential model. The only unusual thing is returning a value from the lambda layer which is also the input to the network. This value is necessary to calculate the value of the loss function, and getting it in any other clever way breaks the whole learning process.

Network graph — Neural network structure

Code flow#

flowchart LR
QOT((QOTracker))
QOC((QOConstants))
QON((QONetwork))
QOP((QOParams))
tr1["train_generations(...)"]
tr2["train(...)"]
  QOT --> QOC --> QON --> tr1
  QOP --> tr1
  tr1 ---> tr2 ---> tr1