Explaining the advantage of quantum-enhanced physics-informed neural networks

TL;DR

It is shown how quantum computing can improve the ability of physics-informed neural networks to solve partial differential equations to achieve an accurate approximation of the solution in substantially fewer training epochs, particularly for more complex problems.

Abstract

Partial differential equations (PDEs) form the backbone of simulations of many natural phenomena, for example in climate modeling, material science, and even financial markets. The application of physics-informed neural networks to accelerate the solution of PDEs is promising, but not competitive with numerical solvers yet. Here, we show how quantum computing can improve the ability of physics-informed neural networks to solve partial differential equations. For this, we develop hybrid networks consisting of quantum circuits combined with classical layers and systematically test them on various non linear PDEs and boundary conditions in comparison with purely classical networks. We demonstrate that the advantage of using quantum networks lies in their ability to achieve an accurate approximation of the solution in substantially fewer training epochs, particularly for more complex problems. These findings provide the basis for targeted developments of hybrid quantum neural networks with the goal to significantly accelerate numerical modeling.

Figures (16)

• Figure 1: Visualization of the median MSE over 10 training runs as a function of the number of training epochs for networks with 250 parameters trained on 1024 training points, for a PDE defined by $L=0.1, N=1.0$ with $u_t^{xsin}$ and $F^{xsin}$ for boundary condition and forcing. The epoch ratio to reach a certain MSE (right) and MSE ratio per epoch (top) are also shown. MSEs are calculated every 100 epochs.
• Figure 2: Ratio of epochs needed to reach a given MSE when training cPINNs and qPINNs. The figure presents an extension of the plot on the right in Fig. \ref{['fig:training_example']}, covering all combinations of PDEs, numbers of collocation points, and trainable parameters. Line colors indicate the MSE of the qPINN after $2\cdot 10^4$ epochs, after which training was stopped. The dashed line marks, where cPINNs and qPINNs perform equally.
• Figure 3: Ratio of MSEs for the cPINN and qPINN as function of number of epochs trained (eps). The figure presents a summary of the plot on the top in Fig. \ref{['fig:training_example']}, covering all combinations of PDEs, numbers of collocation points, and trainable parameters. Line colors indicate the final MSE of the qPINN for the corresponding PDE. The dashed line marks, where cPINNs and qPINNs perform equally.
• Figure 4: Success ratio of cPINNs for various numbers of collocation points and types of boundary conditions and forcings (Eqs. \ref{['eq:xsin_1']}, \ref{['eq:xsin_2']} and Eqs. \ref{['eq:poly_1']}, \ref{['eq:poly_2']}). Trainings are considered not successful when the network gets stuck in a local minimum, as visualized by the figure in the bottom right corner. When the number of collocation points is sufficiently high the training is always successful. Even though the training of qPINNs can be slowed down by reduction of training data, did not get stuck for any of the tests, as illustrated by the orange line.
• Figure 5: Schematic representation of a hybrid neural network architecture with $n_q$ qubits used as qPINN. The spatial coordinate $x$ and time $t$ are input to a classical dense neural network (left block) depending on a set of parameters $\boldsymbol{\boldsymbol{\theta}}$. The outputs of the classical network $i_j$ are encoded into a quantum circuit (center block) through alternating rotational gates $R_{Y/X}(i_j)$ around the $Y$ and $X$ axes. Then, parametrized single-qubit operations followed by entangling operations, both represented by $V(\boldsymbol{\boldsymbol{\theta}})$, are applied. These encoding and parametrized layer sequences (marked in purple) can be repeated multiple times to increase expressivity. Expectation values of the Pauli-Z operator are measured for each qubit, and the resulting quantum outputs $o_j$ are fed into a second classical network to produce the final output $u$ (right block).