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CLNO: Conservative Local Neural Operator for Compressible Flow Prediction
Buxuan Wang, Rongzu Li, Qingsong Zou
TL;DR
High-resolution prediction of unsteady compressible flows can be costly, while local-state evolution must retain fine-scale information and finite-volume conservation. CLNO combines conservative local evolution, fine-scale reconstruction, and Reynolds–Mach conditioning; the reported experiments show improved accuracy and generalization over LNO across resolutions, domains, and boundary conditions.
Problem
High-resolution compressible-flow simulations are costly, and local mean states alone cannot represent internal gradients, waves, and vortical structures that influence flow dynamics.
Source: §1 · pp. 1–2Method
CLNO evolves subdomain-integrated conservative quantities with shared interface fluxes, reconstructs fine-scale fields with zero-mean basis functions, and conditions its networks on Reynolds and Mach numbers.
Source: §1 · pp. 2–3Results
CLNO improves prediction accuracy and generalization over LNO across spatial resolutions, computational domains, and boundary conditions, with improved long-time prediction and transfer to unseen parameter combinations without retraining.
Source: §5 · p. 16Takeaways & Limitations
CLNO provides a conservative and generalizable framework for compressible-flow prediction through local conservative evolution, fine-scale reconstruction, and physical-parameter modulation.
Source: §5 · p. 16CLNO preserves integral conservation in each local subdomain, but the reconstructed field is not explicitly constrained to satisfy the governing equations pointwise.
Source: §5 · p. 16
Abstract
from arXiv · showhide
We propose a conservative local neural operator (CLNO) for accurate and generalizable autoregressive prediction of compressible flows. Building upon the locality and translation invariance of the local neural operator (LNO), CLNO introduces a conservative evolution framework based on local subdomains of fixed physical size, separating local mean-state evolution from fine-scale reconstruction. The flow is encoded into subdomain-integrated conservative states and fine-scale features, while a shared local network predicts interface fluxes for finite-volume updates that enforce discrete conservation. A convolutional reconstruction network recovers fine-scale structures using zero-mean basis functions, preserving the updated local means and providing information for subsequent evolution. To accommodate different physical parameters, Reynolds--Mach number conditioning modulates the encoding, flux-prediction, and reconstruction networks, enabling a single model to learn parameter-dependent dynamics and generalize to unseen parameter combinations without retraining. CLNO supports prediction across spatial resolutions, computational domain sizes, and boundary conditions. Numerical experiments demonstrate improved generalization and competitive or higher accuracy compared with LNO, including order-of-magnitude error reductions in representative cases, at approximately three to five times the inference cost on periodic benchmarks. These results demonstrate accurate, generalizable, and physically consistent flow prediction through conservative local evolution and fine-scale reconstruction.
1. Introduction
CLNO targets accurate, generalizable compressible-flow prediction while addressing the cost of repeated high-resolution simulations and the lack of discrete conservation guarantees in LNO. It combines local conservative evolution, fine-scale reconstruction, and Reynolds–Mach conditioning.
1. Introduction
- Unlike LNO's direct next-state prediction, CLNO targets discrete conservation of mass, momentum, and total energy during local time evolution.LNO's grid-point-defined receptive range also changes in physical extent with spatial resolution, and truncated local Legendre representations may limit fine-scale recovery.Source: §1 · pp. 1–2
- CLNO uses fixed-physical-size local subdomains to support transfer across physical-grid resolutions while reconstructing unresolved gradients, waves, and vortical structures without changing local conservative states.The introduction identifies high-resolution simulation costs, especially for long-time prediction and repeated calculations, as motivation for efficient, generalizable methods retaining essential physical properties.Source: §1 · pp. 1–2
- The method evolves subdomain-integrated conservative quantities through shared interface fluxes, then reconstructs fine-scale structures with zero-mean Legendre basis functions.The reconstruction preserves consistency with the updated local conservative states.Source: §1 · pp. 2–3
- CLNO improves accuracy and generalization over LNO across evaluated resolutions, domains, and boundary conditions, with order-of-magnitude error reductions in representative cases.The authors also report improved long-time prediction and transfer to unseen parameter combinations without retraining.Source: §1 · p. 3
- Reynolds–Mach conditioning modulates encoding, flux prediction, and reconstruction so one model can represent different flow regimes and generalize to unseen parameter combinations without retraining.The motivation is that different Reynolds–Mach combinations generally yield different evolution, making a separate operator for each combination inefficient.Source: §1 · pp. 2–3
2. Problem formulation
The section formulates compressible Navier–Stokes evolution in conservative form and seeks a shared neural operator that transfers across grids, domains, boundary conditions, and physical parameters. Its local updates are constrained by finite-volume conservation on fixed-physical-scale subdomains.
2. Problem formulation
- The governing compressible Navier–Stokes equations are posed in conservative form, with convective and viscous fluxes dependent on the conservative state and physical parameters.The state comprises density, momentum components, and total energy; the parameter pair is Reynolds and Mach numbers.Source: §2 · p. 3
- Local conservation is imposed by partitioning the domain into nonoverlapping subdomains of prescribed physical scale and requiring finite-volume conservation on that partition.The integral form follows by integrating the governing equations over each subdomain and applying the divergence theorem.Source: §2 · pp. 3–4
- The target is a shared evolution mapping across initial conditions, resolutions, domains, boundary conditions, and Reynolds–Mach combinations, with direct inference at unseen parameter combinations.The physical parameters modulate the local networks, while the domain partition and boundary treatment depend on the computational domain and boundary conditions.Source: §2 · p. 3
- Each internal face uses one shared time-averaged numerical flux with opposite orientations in adjacent subdomains, enforcing conservative exchange between them.The stated finite-volume law uses face measures and orientation signs to sum flux contributions for each subdomain.Source: §2 · p. 4
3. Method
CLNO evolves conservative local means through shared interface fluxes and reconstructs fine-scale flow with zero-mean basis functions. Its encoding, flux prediction, and reconstruction are conditioned on Reynolds and Mach numbers, while domain geometry and boundary conditions determine the realization.
3.1. Overall framework
- Reynolds and Mach numbers condition all three modules; the computational domain and boundary conditions determine their geometric and boundary realization, with trainable parameters shared.The autoregressive framework encodes the current flow, advances local states, reconstructs the predicted field, and re-encodes it for the next step.Source: §3.1 · pp. 4–5
3.2. Local encoding module
- Each local subdomain is encoded using its mean conservative state and learned fine-scale features extracted from grid points in a fixed physical neighborhood.Graph edges connect points within a fixed physical distance, and the encoding produces local states for all subdomains.Source: §3.2 · pp. 5–6
3.3. Flux prediction module
- A shared predicted flux enters adjacent subdomain updates with opposite signs, so internal flux contributions cancel in the discrete global balance.A convolutional network predicts interface fluxes, which advance local integral conservative states through a finite-volume scheme.Source: §3.3 · p. 6
3.4. Local reconstruction module
- Zero-mean Legendre reconstruction preserves the updated integral conservative states independently of the predicted coefficients, and the reconstructed flow is re-encoded for the next step.A convolutional network predicts the coefficients of the local Legendre expansion.Source: §3.4 · p. 7
3.5. Boundary treatment
- Boundary conditions enter through interface fluxes; periodic boundaries pair opposite interfaces as internal interfaces with one shared flux and opposite orientations.For characteristic boundaries, incoming convective information is prescribed and outgoing information comes from the interior; near nonperiodic boundaries, one-sided or masked stencils are used.Source: §3.5 · pp. 7–8
4. Experiments
The experiments evaluate autoregressive compressible-flow forecasts across dimensions, spatial configurations, boundary conditions, and unseen Reynolds–Mach combinations. CLNO generally outperforms LNO, including on spatial-transfer and parameter-generalization tests, while ablations show contributions from both conservative flux evolution and fine-scale reconstruction.
4. Experiments
- The evaluation covers prediction accuracy, transfer across resolutions, domain sizes and boundary conditions, unseen physical parameters, and the contributions of conservative evolution and fine-scale reconstruction.Fixed-parameter benchmarks address accuracy and spatial transfer; separate experiments assess parameter generalization and component contributions.Source: §4 · p. 8
4.1. Benchmarks and evaluation protocol
- The fixed-parameter benchmarks use periodic training data and autoregressive forecasts from the same initial conservative state; one- and two-dimensional standard grids are 1024 cells and 64 × 64 cells.The one-dimensional evaluation includes periodic waves, reflecting obstacles, contact discontinuities, and a pressure-jump problem across training, refined, and extended-domain configurations.Source: §4.1 · p. 8, §4.2 · p. 9
4.2. One-dimensional compressible Navier–Stokes equations
- Across the three tested one-dimensional spatial configurations, CLNO has the lowest periodic-wave trajectory error; on the refined grid, errors are 1.346% for CLNO and 27.989% for LNO.For nonperiodic contact transport, CLNO stays below 1% in all three configurations while LNO exceeds 19%; on the refined obstacle test, errors are 2.993% and 26.103%, respectively. Across the original 28 one-dimensional tests, CLNO has lower full-trajectory error than LNO.Source: §4.2 · pp. 9–11
4.3. Two-dimensional compressible Navier–Stokes equations
- 3.306% mean trajectory error on held-out two-dimensional vortical flows, versus 15.901% for LNO; CLNO also has the lowest error on the high-wavenumber test.For the representative held-out flow, instantaneous conservative-state errors at t=8 are 2.382% for CLNO and 22.163% for LNO.Source: §4.3 · pp. 11–12, Table 2 · p. 12
4.4. Generalization across Reynolds and Mach numbers
- Across ten initial fields and 64 forecast frames, CLNO’s mean trajectory errors are 0.140% at the held-out pair (7500, 0.45) and 0.511% at the outside-range pair (5000, 0.50), lower than all three comparison models.The outside-range case has higher error, but the mean remains below 1% in both evaluated unseen parameter settings.Source: §4.4 · p. 14
4.5. Ablation studies
- On two selected held-out two-dimensional trajectories, the full model has the lowest error; removing flux evolution raises trajectory error from 1.384% to 1.953% and conservation error to 0.337%.Suppressing fine-scale reconstruction raises trajectory error to 3.351% while retaining a conservative latent update, and is faster because it omits the fine-grid decoder.Source: §4.5 · p. 15, Table 4 · p. 16
5. Discussion and conclusion
CLNO combines conservative local evolution, fine-scale reconstruction, and Reynolds–Mach conditioning to predict compressible flows across settings. The conclusion identifies the absence of an explicit pointwise governing-equation constraint on reconstructed fields as an area for further work.
5. Discussion and conclusion
- CLNO combines local conservative evolution, zero-mean Legendre reconstruction, and Reynolds–Mach conditioning to predict parameter-dependent flow evolution, including unseen parameter combinations without retraining.The conclusion also reports improved accuracy and generalization over LNO across spatial resolutions, domains, and boundary conditions.Source: §5 · p. 16
- Reconstructed fields are not explicitly constrained to satisfy the governing equations pointwise, despite CLNO preserving integral conservation for each local subdomain.The discussion suggests local physics-informed losses or differentiable physical corrections as possible improvements.Source: §5 · p. 16
- CLNO is presented as a conservative, generalizable framework and a foundation for adaptive, physically consistent neural operators.The proposed future directions include adapting local-subdomain size and dynamically controlling active Legendre modes.Source: §5 · p. 16
A. Experimental data and implementation details
The appendix records the numerical reference data, model configurations, and rollout protocols used in the experiments. Fixed-parameter and Reynolds–Mach studies use separate datasets and checkpoints, and parameter values are prescribed rather than recomputed from fields or domains.
A. Experimental data and implementation details
- The appendix documents the numerical reference data, model configurations, and rollout protocols underlying Section 4.This passage introduces the appendix's scope rather than specifying the individual protocols.Source: A. Experimental data and… · p. 17
- Fixed-parameter and Reynolds–Mach experiments use separate datasets and checkpoints.Source: A. Experimental data and… · p. 17
- Reynolds and Mach numbers are prescribed dimensionless equation parameters, not recomputed from an initial field or changed computational domain.Source: A. Experimental data and… · p. 17
A.1. Fixed-parameter one-dimensional data
The one-dimensional appendix specifies the training reference, independent test references, spatial-transfer configurations, and initial-field and boundary protocols. It also provides profile and space–time visualizations for the periodic wave, obstacle, and pressure-discontinuity cases.
A.1. Fixed-parameter one-dimensional data
- The periodic 1D training reference uses 124 trajectories of 201 frames on a 1024-cell domain, split into 108 training, eight validation, and eight held-out sequences.The solver uses fifth-order WENO reconstruction, Lax–Friedrichs flux splitting, physical viscous stress and Fourier heat conduction, and third-order SSP Runge–Kutta integration.Source: §A.1 · p. 17
- Independent obstacle, contact, and pressure-jump references use a finite-volume Rusanov flux, centered Navier–Stokes diffusion, and third-order SSP Runge–Kutta stepping.They are sampled at Δt=0.01, with CFL and viscous substeps within each stored-frame interval.Source: §A.1 · p. 17
- The three spatial configurations use (N, L)=(1024, 1), (1536, 1), and (1536, 1.5); the CLNO latent-cell width remains 1/128.Grid refinement changes cell count at fixed domain length, while domain extension retains grid spacing; the latent-cell counts are 128, 128, and 192.Source: §A.1 · p. 17
- The periodic-wave and obstacle tests share a Fourier initial field with periodic outer boundaries; the plotted realization uses seed 20261026.The periodic-wave and obstacle initial fields have initial rms Mach number 0.089, while the contact field has 0.181.Source: §A.1 · pp. 17–18
- The temporary reflecting obstacle occupies x/L∈[0.4375, 0.5625] for 0.2≤t<0.8, with periodic exterior boundaries; the domain is otherwise fluid.Figure 11 shows space–time density, velocity, and pressure with the active interval shaded; Figure 12 shows profiles as the obstacle appears and is removed.Source: §A.1 · p. 18, Figure 11 · p. 19, Figure 12 · p. 19
- Independent 1D forecasts contain 100 future frames; the pressure-jump test uses constant-extrapolation outer boundaries.Figure 7? No figure number is assigned here; Figure 4? The supplied pressure-discontinuity profiles and space–time comparison are identified as Figures 13 and 14.Source: §A.1 · p. 19, Figure 13 · p. 20, Figure 14 · p. 20
A.2. Fixed-parameter two-dimensional data
The two-dimensional appendix specifies the periodic training dataset, held-out and high-wavenumber initial fields, and a matched spatial-transfer experiment. The supplied figure compares the reference, LNO, and CLNO for a standard-grid vortical flow.
A.2. Fixed-parameter two-dimensional data
- The 2D training data comprise 100 periodic trajectories on a 64×64 grid, each with 161 frames over t∈[0,8], split into 80 training, 10 validation, and 10 held-out trajectories.Training states contain three to five periodic vortices and smooth Fourier perturbations in density and pressure.Source: §A.2 · p. 20
- The independent high-wavenumber test raises the maximum Fourier wavenumber to eight, compared with at most four in training perturbations.The high-wavenumber field uses seed 20261006.Source: §A.2 · pp. 20–21
- Matched transfer uses (N, L)=(64, 1), (96, 1), and (96, 1.5), with unchanged equation parameters and 16×16 or 24×24 CLNO latent cells.The extended-domain case uses 24×24 cells at the same latent-cell width; forecasts contain 20 frames at Δt=0.05.Source: §A.2 · p. 20
- Held-out and matched spatial-transfer cases use periodic vortical fields with three to five vortices, randomly sampled positions, radii, strengths, and mean velocity.The vortical cases use Fourier fields with K=4; the matched configurations share one normalized-coordinate initial field and seed.Source: §A.2 · pp. 20–21
- Figure 15 compares reference, LNO, CLNO, and absolute-error fields for matched vortical flow on the standard 64×64 grid at t=0.75.Source: Figure 15 · p. 21
A.3. Neural architecture configurations
The reported CLNO architectures combine local message and flux-prediction blocks with convolutional fine-scale reconstruction. Parameter conditioning adds FiLM mappings at encoding, flux prediction, and reconstruction while retaining the dimension-matched fixed-parameter design.
A.3. Neural architecture configurations
- CLNO uses three latent message blocks and a direct interface-flux head with a 1-point lift, depthwise 3-point convolution, pointwise mixing, and final 3-point convolution.The reconstruction head uses two 3-point convolutions, yielding an effective stencil width of five.Source: §A.3 · pp. 21–22
- Table 5 reports fixed-parameter architecture settings; its notation identifies LNO local-resolution parameters, FNO modes per spatial direction, and DeepONet branch–trunk contraction rank.Table 6, by contrast, reports training settings for the reported checkpoints.Source: Table 5 · p. 22, Table 6 · p. 22
- The 1D decoder uses four nonconstant Legendre modes per conservative variable; the 2D decoder uses 24 nonconstant tensor-product modes plus scalar amplitude gates for density and internal energy.Both CLNO checkpoints use two internal substeps per stored frame.Source: §A.3 · p. 22
- Parameter-conditioned CLNO retains dimension-matched fixed-parameter widths, latent geometry, and heads, adding FiLM mappings at encoding, flux prediction, and reconstruction.The 1D FiLM checkpoint is evaluated with two internal substeps per frame.Source: §A.3 · p. 22
A.4. Optimization and inference settings
The appendix specifies training curricula, ablation variants, and rollout conventions for the reported comparisons. CLNO advances with two internal substeps per stored frame, while the other compared operators use one learned map per frame.
A.4. Optimization and inference settings
- For the 64 × 64 Navier–Stokes benchmark, CLNO and LNO share a 120-epoch curriculum and trajectory split, while 1D LNO follows a separately trained official baseline protocol.Checkpoints were selected by validation error, and reported epoch counts are minimum completed lengths.Source: §A.4 · p. 22
- The 1D CLNO curriculum progresses through 1, 2, 4, 8, and 10 frames; the 2D fixed-parameter curriculum progresses from 1 to 64 frames.CLNO uses AdamW with weight decay 10^-5 and cosine decay; the 2D phases use ten epochs except 20 at 64 frames.Source: §A.4 · p. 22
- The flux ablation replaces shared interface-flux divergence with learned per-cell residual updates, while the reconstruction ablation broadcasts each updated latent-cell mean across its physical cells.Both retain remaining full-model components and use the same 64 × 64 training split and rollout curriculum.Source: §A.4 · p. 23
- CLNO divides each stored-frame step Δt into two equal internal substeps, whereas LNO, FNO, and DeepONet apply one learned map per stored frame.Standard and spatial-transfer configurations use the same trained weights; trajectories begin at the reference initial frame and report autoregressively predicted frame counts.Source: §A.4 · p. 23, Table 7 · p. 23
A.5. Parameter-dependent reference data
The parameter study evaluates periodic finite-volume Rusanov reference data across Reynolds–Mach pairs using shared random Fourier initial fields. It holds out one parameter pair and uses a 1024-cell grid without domain transfer.
A.5. Parameter-dependent reference data
- The study uses periodic finite-volume Rusanov references with Newtonian stress and Fourier heat flux, distinct from the fixed-parameter WENO benchmarks.The 1024-cell grid spans [0, 1], and the study does not use domain transfer.Source: §A.5 · p. 23
- Ten pairs combine Re values 5500, 6500, 7500, 8500, and 9500 with Ma values 0.35 and 0.45, reusing ten random Fourier initial fields for 100 sequences.The (7500, 0.45) pair is held out; the other pairs provide 81 training and nine validation sequences.Source: §A.5 · p. 23
- The initial fields reuse the same ten primitive fields for every parameter pair, with Fourier coefficients sampled independently from a standard normal distribution and amplitude scales specified for density, velocity, and pressure.The initial density and pressure are floored at 0.75, and the state and spatial derivatives match periodically at the domain boundaries.Source: §A.5 · p. 23
- All four methods use deterministic CUDA kernels with TF32 disabled in Table 3.Source: §A.5 · p. 23