Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations

• URL: http://arxiv.org/abs/2605.04853v1
• Date: Wed, 06 May 2026 12:50:36 GMT
• Title: Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations
• Abstract summary: HIN-LRI is a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error.<n>A lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close.<n>Experiments show that HIN-LRI improves accuracy over analytical, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains $\mathcal{O}(τ)$, yielding a Gronwall stability factor bounded uniformly in the step size and independent of the spatial resolution. The network is trained end-to-end through a solver-in-the-loop objective that unrolls the full iteration and penalises trajectory error in a Bourgain-type norm, aligning learning with multi-step solver dynamics rather than isolated one-step targets. Under stated assumptions, the global error satisfies $C(\varepsilon_{net}+δ)\,τ^γ\ln(1/τ)$, where $\varepsilon_{net}$ measures the network approximation quality and $δ$ the training shortfall. Experiments on three dispersive benchmarks with rough data show that HIN-LRI improves accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.

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