PaperlayerView exact revision on arXiv

Source-linked AI summary

LSR-Net: Learning the Forward Evolution Operator for Nonlinear Fluid Dynamics

Qian Hou, Sutrisno, Yuqing Li, Zecheng Gan

arXiv:2609.19039v1physics.comp-phcs.LGphysics.flu-dyn

TL;DR

Learning future states from an initial condition when the governing dynamics are unknown is the central problem. LSR-Net combines short-range convolutions with an SOE-based long-range kernel computed in Fourier space. Across three PDE benchmarks, it reports substantially lower RMSE than SR-Net, while long-term wave-equation rollouts degrade as errors accumulate.

  • Problem

    The paper addresses learning a dynamical system's future state from its initial state when the underlying model is unknown.

    Source: §II · p. 1

  • Method

    LSR-Net combines standard short-range convolutions with a learnable SOE long-range kernel, evaluated as a Fourier multiplier.

    Source: §II · pp. 2–3

  • Results

    Across all three PDE benchmarks, LSR-Net achieves substantially lower RMSE than SR-Net; on coupled Burgers, SR-Net reports 0.9999 and LSR-Net 0.0139.

    Source: E. Nonlinear Shallow Water Equations… · p. 5

  • Takeaways & Limitations

    The reported RMSE advantage over SR-Net spans coupled Burgers, spatially varying wave, and shallow-water benchmarks.

    Source: E. Nonlinear Shallow Water Equations… · p. 5, §IV · p. 6

    For the spatially varying wave equation, accumulated errors degrade long-term rollout performance as the horizon increases.

    Source: F. Multi-step inference comparison… · p. 5

Abstract

from arXiv · show

We introduce the Long-Short-Range Neural Network (LSR-Net), a novel neural operator architecture designed for data-driven forward evolution modeling, and extends it to the prediction of nonlinear fluid dynamics. LSR-Net learns the evolution operator of a dynamical system solely from pairs of initial and future state snapshots, which splits the learnable integral kernel into long-range (LR) and short-range (SR) components within stacked network blocks. While the SR component uses standard convolutions to capture local dynamics, the LR component employs a sum-of-exponentials (SOE) representation. This allows for the efficient computation of global interactions as a trainable Fourier multiplier, reducing computational complexity to $O(n \log n)$ where $n$ is the number of pixels in an input snapshot and requiring only a few parameters per channel. LSR-Net is evaluated on three challenging 2D benchmarks: the coupled Burgers equation, the wave equation with a spatially varying coefficient, and the nonlinear shallow water equation {(SWE)}. Results demonstrate that LSR-Net significantly outperforms the baseline short-range network (SR-Net) as well as FNO and DeepONets in predictive accuracy, achieving substantially lower relative errors by effectively capturing both local fine-scale structures and crucial global pattern interactions.

I. Introduction

Operator learning maps initial conditions to future states in function spaces using paired observations, rather than approximating solutions only for fixed parameters. The section presents LSR-Net as a response to long-range dependency challenges in dynamical systems.

I. Introduction

  • Operator learning learns mappings between infinite-dimensional function spaces from paired initial and future snapshots, unlike methods limited to fixed parameters.This frames data-driven forward evolution modeling as learning a system's solution operator from observations.Source: §I · p. 1
  • FNO parameterizes its integral kernel in Fourier space, while DeepONet uses branch and trunk networks to learn operators from data.Both are described as flexible, model-agnostic frameworks for learning complex PDE dynamics.Source: §I · p. 1
  • Existing operator learners can struggle with long-range dependencies, which can lead to inaccurate long-term predictions in pattern-forming systems.Such systems combine global interactions with local fine-scale structures that co-evolve.Source: §I · p. 1
  • LSR-Net splits the learnable kernel into short- and long-range components, using SOE for global interactions and standard convolutions for local dynamics.The section reports superior predictive accuracy across multiple challenging nonlinear fluid dynamics benchmarks.Source: §I · p. 1

II. The Long-Short-Range Neural Network

LSR-Net learns a dynamical system's evolution from its initial state to a future state when the underlying model is unknown. Its stacked blocks combine local convolutions with an SOE-based long-range kernel computed efficiently in Fourier space.

II. The Long-Short-Range Neural Network

  • The model parameters are optimized by minimizing mean squared error between predicted and reference states at time T, and the learned operator can be repeatedly applied for long-term prediction.The evolution-operator task is to predict a future state from the initial state when the underlying model remains unknown.Source: §II · p. 1
  • Each stacked LSR-Net block applies learnable integral kernels and a nonlinear activation, with residual connections to improve training stability.The blocks map the input state at t = 0 to a predicted state at t = T.Source: §II · p. 2, Figure 1 · p. 2
  • The kernel is split into long-range components for global interactions and short-range components for local structure; the latter uses small standard convolutions.Conventional convolutions typically need very large kernels or deep stacks to capture long-range interactions.Source: §II · p. 2
  • The LR kernel uses a learnable SOE representation that assumes translational invariance; higher-dimensional extensions are possible when that symmetry does not hold.The parameters are learned from data without assuming a predefined kernel structure; the representation is motivated by approximation theory for smooth kernels.Source: §II · p. 2
  • Oscillatory LR interactions can be represented with complex exponents, requiring only three trainable parameters per LR channel; s_i = 0 gives the nonoscillatory case.The Fourier multiplier implements the LR convolution using the convolution theorem.Source: §II · pp. 2–3
  • O(n log n) Fourier evaluation replaces O(n^2) direct LR convolution, using an FFT, pointwise multiplier application, and inverse FFT.Here n is the number of pixels in the input snapshot.Source: §II · p. 3

A. Data Generation

The benchmarks use synthetic datasets generated by numerically solving three 2D systems. Each dataset entry represents a trajectory of solution fields over time.

  • The three benchmarks are 2D Burgers, a wave equation with spatially varying coefficient, and a shallow water system used as a nonlinear wave benchmark.The simulations use 128 × 128 uniform periodic grids, with initial conditions sampled from Gaussian random fields with Matérn covariance.Source: A. Data Generation · p. 3
  • Each dataset entry corresponds to a trajectory containing solution fields over time.The number of physical channels is one for Burgers and wave, and three (h, u, v) for shallow water.Source: A. Data Generation · p. 3

B. Model Setting

The model and dataset configurations cover three PDE benchmarks, with benchmark-specific fields and initial conditions. Accuracy is evaluated using relative L2 error.

  • The section summarizes model and dataset settings for the Burgers, spatially varying wave, and nonlinear shallow water benchmarks.Table I is identified as the summary of model and dataset settings across all PDE benchmarks.Source: B. Model Setting · p. 3, Table I · p. 3
  • For Burgers, the 2D system is a coupled vector field (u, v), with periodic boundary conditions and initial conditions sampled from Gaussian random fields.Source: B. Model Setting · p. 3
  • For the wave benchmark, spatially varying wave speed c(x, y) is an additional PINO input alongside an independently sampled initial displacement.Both fields are sampled from Gaussian random fields.Source: B. Model Setting · p. 3
  • For shallow water, the modeled total height has an initial perturbation and zero initial velocity fields; implementations use PyTorch on a single MPS device.The section evaluates accuracy using relative L2 error.Source: B. Model Setting · p. 3

C. 2D Coupled Burgers Equation

On the coupled 2D Burgers benchmark, LSR-Net predicts both velocity components more accurately than SR-Net. The experiment uses periodic, randomly initialized coupled fields and compares predictions at the final time.

  • The Burgers system couples u and v through nonlinear advection and uses viscosity ν = 0.01 on a periodic spatial domain.Initial conditions for both components are independently sampled from Gaussian random fields with correlation length 0.1 and standard deviation 0.2.Source: C. 2D Coupled Burgers Equation · p. 3
  • The dataset contains 500 independent realizations, each stored as a tensor with velocity components, time steps, and two spatial dimensions.Snapshots are recorded every 0.01 time units after solving with a spectral solver at time step 1 × 10^-4.Source: C. 2D Coupled Burgers Equation · p. 3
  • LSR-Net outperforms SR-Net on both coupled velocity components, with more accurate predictions and lower errors.The comparison visualizes each component and shows initial condition, ground truth at t = T, prediction, and absolute error; SR-Net omits the long-range SOE kernel.Source: C. 2D Coupled Burgers Equation · pp. 3–4, Figure 2 · p. 4

D. 2D Wave Equation with Spatially Varying Coefficient

For the wave equation with spatially varying speed, LSR-Net produces more accurate predictions and better captures sharp wave structures than SR-Net. The benchmark uses a scalar wave field and a spatially varying wave speed.

  • The wave speed c(x, y) varies spatially and is sampled from a Gaussian random field, producing heterogeneous propagation dynamics.The initial displacement is independently sampled from another Gaussian random field.Source: D. 2D Wave Equation with Spatially… · p. 4
  • The equation is solved on a periodic domain with time step 10^-4 up to final time T = 5, with snapshots recorded at fixed intervals.Source: D. 2D Wave Equation with Spatially… · p. 4
  • LSR-Net more accurately predicts the wave field and captures sharp structures, while SR-Net has larger errors where spatial variations are strong.The figure compares initial condition, ground truth at t = T, prediction, and absolute error for the two models.Source: D. 2D Wave Equation with Spatially… · p. 4, Figure 3 · p. 4

E. Nonlinear Shallow Water Equations (2D)

On the nonlinear shallow water benchmark, LSR-Net predicts the coupled height and velocity fields more accurately than SR-Net, including velocities that begin at zero.

E. Nonlinear Shallow Water Equations (2D)

  • The SWE prediction challenge combines three coupled variables, zero initial velocities, and nonlinear dynamics.The initial height is a mean value of 1 plus a perturbation, while both initial velocity fields are zero.Source: E. Nonlinear Shallow Water Equations… · pp. 4–5
  • LSR-Net captures the induced velocity dynamics from zero initial velocities and predicts all three SWE variables more accurately than SR-Net.The comparison visualizes η, u, and v for both models alongside their prediction errors.Source: E. Nonlinear Shallow Water Equations… · p. 5
  • RMSE: Burgers SR-Net 0.9999 vs LSR-Net 0.0139; wave SR-Net 0.4297 vs LSR-Net 0.0306; SWE SR-Net 0.4755 vs LSR-Net 0.0181.Table II compares relative RMSE across the three PDE benchmarks.Source: Table II · p. 5, E. Nonlinear Shallow Water Equations… · p. 5

F. Multi-step inference comparison with different methods

The paper compares autoregressive rollouts of LSR-Net, SR-Net, FNO, and DeepONet across three PDE benchmarks. LSR-Net performs best on Burgers and SWE rollouts, while wave-equation errors accumulate for every method.

F. Multi-step inference comparison with different methods

  • Rollouts recursively feed each prediction into the next step, run to 5T, and are evaluated using relative L2 error.The benchmarks are coupled Burgers, wave, and SWE; the rollout interval is T = 0.2 for Burgers and SWE and T = 1 for wave.Source: F. Multi-step inference comparison… · p. 5
  • LSR-Net leads long-term rollouts on Burgers and SWE, while all methods accumulate wave-equation error and LSR-Net leads only through 3T.On Burgers, SR-Net rapidly collapses and DeepONet has larger errors throughout; for SWE, SR-Net and DeepONet show unstable autoregressive behavior.Source: F. Multi-step inference comparison… · p. 5
  • For the wave equation, spatially varying coefficients make the operator non-isotropic, and accumulated errors degrade longer-horizon predictions.LSR-Net has the best short-term accuracy at rollout stages ≤3T.Source: F. Multi-step inference comparison… · p. 5
  • Across the PDE systems, LSR-Net outperforms the ablated SR-Net variant in autoregressive prediction, highlighting the importance of the LR component for stable long-term operator learning.The comparison includes FNO and DeepONet as additional baselines.Source: F. Multi-step inference comparison… · p. 5

IV. Conclusion

The conclusion presents LSR-Net as a model-free forward-evolution approach built from early snapshots and a long-/short-range kernel split. It reports stronger predictive accuracy than SR-Net across three PDE benchmarks, particularly for long-term prediction.

IV. Conclusion

  • LSR-Net is a model-free approach that requires only two sets of early-stage snapshots.Source: §IV · p. 5
  • LSR-Net consistently achieves superior predictive accuracy to SR-Net across all three PDE benchmarks, particularly for long-term predictions.The benchmarks are coupled Burgers, the wave equation with spatially varying wave speed, and the nonlinear shallow water system.Source: §IV · p. 6
  • The long-range module uses an SOE-derived trainable Fourier multiplier for global interactions, with only 3–4 trainable parameters per channel.It is combined with a standard short-range convolutional kernel to learn complex pattern evolution.Source: §IV · p. 6
  • Future work will extend LSR-Net to broader nonlinear systems, including anisotropic systems, irregular geometries, and operators on general manifolds.Source: §IV · p. 6
Loading 2609.19039v1…