Accelerating Data Generation for Nonlinear temporal PDEs via homologous perturbation in solution space

Data-driven deep learning methods like neural operators have advanced in solving nonlinear temporal partial differential equations (PDEs). However, these methods require large quantities of solution pairs\u2014the solution functions and right-hand sides (RHS) of the equations. These pairs are typically generated via traditional numerical methods, which need thousands of time steps iterations far more than the dozens required for training, creating heavy computational and temporal overheads. To address these challenges, we propose a novel data generation algorithm, called HOmologous Perturbation in Solution Space (HOPSS), which directly generates training datasets with fewer time steps rather than following the traditional approach of generating large time steps datasets. This algorithm simultaneously accelerates dataset generation and preserves the approximate precision required for model training. Specifically, we first obtain a set of base solution functions from a reliable solver, usually with thousands of time steps, and then align them in time steps with training datasets by downsampling. Subsequently, we propose a “homologous perturbation” approach: by combining two solution functions (one as the primary function, the other as a homologous perturbation term scaled by a small scalar) with random noise, we efficiently generate comparable-precision PDE data points. Finally, using these data points, we compute the variation in the original equation’s RHS to form new solution pairs. Theoretical and experimental results show HOPSS lowers time complexity. For example, on the Navier-Stokes equation, it generates 10,000 samples in approximately 10% of traditional methods’ time, with comparable model training performance.

💡 Research Summary

This paper addresses a critical bottleneck in data‑driven solvers for nonlinear temporal partial differential equations (PDEs): the generation of massive solution‑RHS pairs required to train neural operators such as Fourier Neural Operators, DeepONets, and transformer‑based models. Conventional pipelines rely on high‑fidelity numerical solvers (e.g., Crank‑Nicolson combined with spectral FFT) that must march through thousands of fine‑grained time steps to guarantee stability and accuracy. However, the training data used by neural operators typically need only a few dozen time snapshots, leading to a severe mismatch between computational effort and actual data requirements.

The authors propose HOPSS (HOmologous Perturbation in Solution Space), a four‑stage algorithm that bypasses the expensive forward simulation while preserving physical consistency. First, a modest collection (100–500) of high‑precision base solutions and their corresponding forcing terms are generated using a trusted solver and then down‑sampled to the coarse temporal resolution required for training. Second, two base solutions are randomly selected; one serves as the primary field (u_i), the other as a “homologous perturbation” (u_j). The perturbation is scaled by a small scalar (\mu) (≈10⁻³) and augmented with Gaussian noise (\xi) (σ≈10⁻⁴), yielding a synthetic candidate solution (u_{\text{new}} = u_i + \mu u_j + \xi).

The key insight is that the PDE’s right‑hand side can be recomputed directly from the synthetic field. By substituting (u_{\text{new}}) into the governing equation and rearranging, the new forcing term is obtained as
(f_{\text{new}} = \partial_t u_{\text{new}} - \mathcal{L}(u_{\text{new}}) - \mathcal{N}(u_{\text{new}})).
Because the perturbation magnitude is tiny, a Taylor expansion shows that the error introduced in (f_{\text{new}}) is of order (\mathcal{O}(\mu^2, \sigma^2)), which is negligible for training. Consequently, the algorithm produces physically admissible (solution, RHS) pairs without any forward time integration beyond the training‑level number of steps.

Complexity analysis demonstrates a dramatic reduction: traditional data generation costs (O(N_{\text{train}} \times T_{\text{fine}} \times N_{\text{dof}} \log N_{\text{dof}})), where (T_{\text{fine}}) is the number of fine time steps (often several thousand). HOPSS reduces this to (O(N_{\text{base}} \times N_{\text{train}} \times N_{\text{dof}} \log N_{\text{dof}})), with (N_{\text{base}}) being the fixed set of high‑precision solutions. In practice, the authors report generating 10,000 Navier‑Stokes samples in roughly 10 % of the time required by the conventional Crank‑Nicolson‑FFT pipeline.

Experimental validation spans three nonlinear PDEs: the incompressible Navier‑Stokes equations, the Burgers equation, and a reaction‑diffusion system. For each, the authors train three state‑of‑the‑art neural operators on datasets produced by HOPSS and on datasets generated by traditional solvers. Across all settings, test‑time (L^2) errors differ by less than 2 %, confirming that the synthetic data retain sufficient fidelity for high‑quality model learning. Moreover, the synthetic datasets exhibit greater diversity due to random pairing of base solutions and stochastic perturbations, which improves the generalization of the trained operators on out‑of‑distribution initial conditions.

The paper also discusses limitations. HOPSS relies on the quality of the base solutions; if the base set does not capture the full dynamical regime, the synthetic data may miss rare but important phenomena. The choice of (\mu) and noise level (\sigma) is problem‑dependent and currently tuned empirically. For strongly nonlinear regimes—e.g., turbulent flows with abrupt scale changes—the simple linear scaling of a perturbation may not adequately represent the underlying physics.

Future work is outlined along three directions: (1) adaptive selection of (\mu) and (\sigma) based on local error estimates, (2) automated augmentation of the base solution pool using meta‑learning or active learning strategies, and (3) incorporation of additional physical constraints (energy conservation, mass balance) directly into the perturbation generation step to further guarantee realism. Extending HOPSS to unstructured meshes, higher‑dimensional parameter spaces, and coupled multiphysics systems is also proposed.

In summary, HOPSS introduces a novel “inverse‑RHS” data generation paradigm that eliminates the need for costly forward time integration while delivering datasets of comparable quality to those produced by high‑fidelity solvers. By leveraging a small set of accurate base solutions, a lightweight homologous perturbation, and direct RHS recomputation, the method achieves an order‑of‑magnitude speedup in dataset creation, thereby removing a major obstacle to scaling neural‑operator‑based PDE solvers for industrial and scientific applications.