Quasi-Random Physics-informed Neural Networks

Physics-informed neural networks have shown promise in solving partial differential equations (PDEs) by integrating physical constraints into neural network training, but their performance is sensitive to the sampling of points. Based on the impressive performance of quasi Monte-Carlo methods in high dimensional problems, this paper proposes Quasi-Random Physics-Informed Neural Networks (QRPINNs), which use low-discrepancy sequences for sampling instead of random points directly from the domain. Theoretically, QRPINNs have been proven to have a better convergence rate than PINNs. Empirically, experiments demonstrate that QRPINNs significantly outperform PINNs and some representative adaptive sampling methods, especially in high-dimensional PDEs. Furthermore, combining QRPINNs with adaptive sampling can further improve the performance.

💡 Research Summary

**
The paper introduces Quasi‑Random Physics‑Informed Neural Networks (QRPINNs), a new sampling strategy for Physics‑Informed Neural Networks (PINNs) that replaces the conventional Monte‑Carlo (MC) point selection with low‑discrepancy sequences from Quasi‑Monte‑Carlo (QMC) methods. The authors begin by reviewing the standard PINN formulation, where the loss consists of residual, initial‑condition, and boundary‑condition terms that are approximated by MC integration over randomly drawn collocation points. They prove (Theorem 1) that, under smoothness assumptions on the true solution, the overall error of a PINN is dominated by the quadrature error of the MC approximation.

QMC methods, which use deterministic sequences such as Halton or Sobol’, achieve a convergence rate of O(N^{-(1‑ε)}) with ε∈(0,1), substantially faster than the O(N^{-1/2}) rate of MC. By inserting QMC points into the PINN loss, the authors derive a new error bound showing that QRPINNs inherit the superior QMC convergence, leading to a theoretical improvement in the PINN error rate.

A practical issue is that pure QMC is fully deterministic, eliminating the stochasticity that modern stochastic gradient descent (SGD) relies on. To retain randomness while preserving low discrepancy, the authors propose Randomized QMC (RQMC): the full low‑discrepancy set is generated once, and at each training epoch a random subset of size N is sampled. Theorem 2 quantifies the expected error of RQMC as a combination of the original QMC error and terms depending on the sampling ratio k = N/N_total, showing that when k is not too small the RQMC error remains close to the QMC error.

Empirical evaluation focuses on high‑dimensional integration tests (sin and exponential functions) and several high‑dimensional PDEs (wave, heat, and nonlinear reaction‑diffusion equations). In the integration tests, QMC consistently outperforms MC, and the advantage grows with dimension d (up to d = 100). For the PDE experiments, QRPINNs achieve markedly lower L2 errors than standard PINNs using MC points, and also surpass a suite of recent adaptive sampling strategies—including RAD, RANG, FI‑PINNs, and AAS—by one to two orders of magnitude in error reduction. Moreover, when adaptive sampling is combined with QRPINNs, further improvements of 10–30 % are observed, indicating that the two ideas are complementary.

The authors discuss limitations: (i) in low‑dimensional problems the benefit of QMC diminishes; (ii) the choice of low‑discrepancy sequence influences practical performance; (iii) the theoretical rates assume sufficiently expressive neural networks and appropriate hyper‑parameters (learning rate, batch size, network width/depth). They suggest future work on (a) exploring alternative sequences (e.g., Niederreiter‑Xing, Korobov) and their randomized variants; (b) integrating dimension‑wise importance weighting to create hybrid QMC‑adaptive schemes; and (c) scaling RQMC‑based training to distributed environments.

In summary, QRPINNs demonstrate that leveraging quasi‑random low‑discrepancy sampling can fundamentally accelerate the convergence of physics‑informed neural solvers, especially for high‑dimensional PDEs, and that the approach can be further enhanced by existing adaptive sampling techniques. This work highlights sampling strategy as a critical lever for the success of deep learning‑based scientific computing.