Operator Learning with Neural Fields: Tackling PDEs on General Geometries

1 Introduction

• Existing neural operators such as DeepONet and FNO still rely on fixed grids during training and inference, limiting use with irregular sampling or new geometries.An FNO variant for more general geometries is described as focused on design tasks.Source: §1 · pp. 1–2
• CORAL encodes functions in compact, low-dimensional latent spaces and maps between their representations, supporting PDE solving, spatio-temporal forecasting, and design problems.The approach leverages implicit neural representations and is presented as flexible across problem domains.Source: §1 · p. 2
• CORAL learns mappings from functions sampled on irregular meshes and maintains consistent performance on grids unseen during training.The authors present this as suited to complex geometries and non-uniform grids.Source: §1 · p. 2
• CORAL demonstrates state-of-the-art performance across representative physical-modeling tasks and geometries, including convex and non-convex domains and planar and spherical surfaces.The tasks include initial value problems, geometry-aware inference, dynamics modeling, and forecasting.Source: §1 · p. 2
• CORAL uses compact latent codes to support fast inference, unlike methods that operate directly in higher-dimensional function representations.The paper presents this efficiency as a feature of its latent representation.Source: §1 · p. 2

Mesh-based networks for physics

• CNNs expect regular grids, while GNNs handle irregular meshes but can inherit biases from training mesh neighborhoods and require substantial memory for those neighborhoods.The paper notes that mesh bias can hinder generalization to different node locations or sparsity levels.Source: §2 · p. 2

Operator learning

• DeepONet requires the same predefined observation locations across training and testing, whereas FNO cannot use irregular grids because it relies on the fast Fourier transform.The cited flexible-geometry FNO extension is tailored to design problems.Source: §2 · p. 2

Spatial INRs

• Spatial INRs model data as implicit functions of coordinates and can be queried at any location, but a basic INR encodes only one sample or function.Prior work uses meta-learning, auto-encoders, or modulation to decode multiple functions with per-sample parameters.Source: §2 · p. 3
• Prior INR applications include forecasting and reduced-order modeling, but the forecasting approach is task-specific and described as more computationally costly than CORAL.The paper distinguishes its data-driven approach from work that informs an INR with known PDEs.Source: §2 · p. 3

3.1 Problem Description

• For each training example, CORAL targets input-output function pairs sampled on free-form spatial grids, with separate training and testing grids for dynamics modeling.Initial-value and geometry-aware examples may each use a specific grid; dynamics modeling uses one grid for training examples and another for testing.Source: §3.1 · pp. 3–4

3.2 Model

• CORAL represents input and output functions with modulated INRs and learns mappings between their low-dimensional latent representations.Shared INR parameters represent multiple functions, while function-specific modulation parameters are generated from latent codes by hypernetworks.Source: §3.2 · p. 4
• The framework encodes an input function into a latent code, transforms that code with a task-dependent model, and decodes it into an output function.The processor can be an MLP or, for dynamics modeling, a neural ODE solver; the decoded INR can be queried at any spatial coordinate.Source: §3.2 · pp. 4–5
• Input encoding uses auto-decoding to find a compact latent code whose modulated INR reconstructs the observed input function.The code is optimized by gradient descent against observations on the example's grid; because the INR can be queried anywhere, encoding does not require a fixed mesh.Source: §3.2 · pp. 4–5

3.3 Practical implementation: decoding by INR Modulation

• CORAL uses SIREN as its coordinate-based INR backbone and applies shift modulations to reduce modulation-space size and computational cost.SIREN uses sine activations with a specific initialization scheme.Source: §3.3 · p. 5

3.4 Training

• Training first fits the modulated INRs, then trains the forecast model in the low-dimensional code space; this is reported as faster and more stable than end-to-end training with similar performance.The method limits encoding steps to discourage memorization and uses second-order meta-learning based on CAVIA; three inner steps were sufficient for low reconstruction errors in most applications.Source: §3.4 · pp. 5–6

4.1 Initial Value Problem

• The initial-value experiments use non-convex flow domains with obstacles and evaluate both full meshes and meshes down-sampled to 20% of their nodes.Training uses all mesh points; the datasets include Cylinder and Airfoil, with different meshes for each example.Source: §4.1 · p. 6
• In the Full setting, CORAL is best on Cylinder and second on Airfoil; under Sparse evaluation, it gives the lowest-error predictions on the full mesh.Sparse evaluation is more challenging for Cylinder than Airfoil given their average mesh sizes; GraphSAGE generalizes worse than the local model, while MP-PDE accounts for node distances and feature differences.Source: Table 1 · p. 7, §4.1 · p. 7

4.2 Dynamics Modeling

• For dynamics forecasting, CORAL encodes the initial state and autoregressively predicts successive states in latent space, then decodes them as functions.The approach uses a Neural ODE solver, allowing forecasts at arbitrary or irregular time steps; experiments use RK4 and a single modulated INR across each sequence.Source: §4.2 · p. 7, §4.2.1 · p. 7

4.2.1 Implementation with Neural ODE

• Training first fits the INR to auto-decode trajectory states, then learns the Neural ODE processor over fixed latent codes.The two stages are separated, and the solver operates on the low-dimensional code embedding space.Source: §4.2.1 · p. 7

4.2.2 Experiment details

• Dynamics experiments use Navier–Stokes and spherical shallow-water trajectories, with temporal extrapolation, changed spatial subsampling grids, and additional upsampling evaluation.The stated protocols split 40 timestamps into In-t and Out-t, test forecasting through t = 39, and evaluate on separate train and test grids with the same sparsity.Source: §4.2.2 · p. 8

4.2.3 Results

• CORAL performs strongly across temporal and spatial scenarios on both dynamics datasets; only DINo shows similar stability across subsampling and extrapolation, while lower sampling and longer horizons reduce performance.Because models train only on In-t, errors accumulate during Out-t evaluation.Source: §4.2.3 · p. 8, Table 2 · p. 9

4.3 Geometry-aware inference

• Geometry-aware inference maps a deformation of a reference grid to a physical field and targets generalization to new geometries such as airfoil shapes.The benchmarks cover airfoil flow, pipe flow, and hyper-elastic materials, with Geo-FNO and Factorized-FNO as main baselines.Source: §4.3 · p. 9
• CORAL achieves the lowest relative error on Airfoil and Elasticity, but is slightly below Factorized-FNO and Geo-FNO on Pipe.The paper also applies CORAL to airfoil design by optimizing geometry to minimize the drag-to-lift ratio.Source: Table 3 · p. 9, §4.3 · p. 10

• CORAL inherits INR limits in training time and representation power; it is slower to train than operator methods and has not yet been evaluated on large practical problems.The paper notes that some physical phenomena may not be representable by INRs.Source: §5 · p. 10

• CORAL removes constraints on the input-output mesh and handles spatial sampling or geometry variations across a wide range of scenarios.Compact latent codes enable efficient, fast inference in a condensed representation space.Source: §6 · p. 10

A.1 Initial Value Problem

• Cylinder and Airfoil initial-value inputs and outputs use the first and last frames of trajectories from irregular triangular-mesh CFD simulations.Cylinder uses incompressible flow with 600 timestamps at 0.01 s intervals; Airfoil uses compressible flow with 600 timestamps at 0.008 s intervals.Source: §A.1 · p. 14

A.2 Dynamics Modeling

• Dynamics modeling covers 2D Navier–Stokes trajectories and spherical shallow-water dynamics, with task-specific grids, time windows, and train/test splits.Navier–Stokes uses 256 training and 16 evaluation trajectories; shallow-water trajectories are formed from 16 training and 2 test simulations, then sliced into 64 and 8 sub-trajectories.Source: §A.2 · pp. 14–15

A.3 Geometric aware inference

• Geometry-aware inference evaluates NACA-Euler airfoil flow, elasticity with variable void shapes, and flow in pipes with variable centerlines.The setups include solver-generated mesh inputs and outputs, with geometric variation specified through airfoil control-node displacements, void-radius priors, or pipe centerline control nodes.Source: §A.3 · pp. 15–16

B Implementation Details

• The experiments use PyTorch, and the reported development and experiment computation time is approximately 400 days.Source: B Implementation Details · p. 16

B.1.1 Architecture Details

• CORAL’s decoder uses shift-modulated SIREN; the authors report that adding scale modulation did not improve performance and that shift-only modulation trained more stably.The processor is an MLP with skip connections and Swish activations; input encoding starts from a zero code and uses gradient descent to minimize reconstruction loss.Source: §B.1.1 · p. 16

B.1.2 Training Details

• Training fits the modulated input and output INRs first, then fixes their latent codes and trains the processor to map input codes to output codes.The encoding phases use a few gradient steps, and their gradients are back-propagated through those inner steps to update INR and hypernetwork parameters.Source: §B.1.2 · pp. 16, 18, Figure 5 · p. 18
• CORAL uses task-dependent code normalization, including shared input/output statistics for Cylinder and dynamics modeling, separate statistics for Airfoil, and input-only normalization for geometry-aware inference.Codes are normalized by subtracting the mean and dividing by the standard deviation.Source: §B.1.2 · pp. 16–17

B.1.3 Inference Details

• At inference, CORAL encodes a new input using the training-time number of inner steps, normalizes its code for the processor, then decodes the processed code.The output encoder is used during training but not at inference; the described inference uses three inner steps for all tasks.Source: §B.1.3 · p. 18, §B.1.2 · p. 18

B.1.4 Choice of Hyperparameters

• Hyperparameter tables specify CORAL settings for initial-value and geometry-aware inference tasks and for dynamics-modeling tasks.Source: §B.1.4 · p. 19, Table 4 · p. 20, Table 5 · p. 20
• For dynamics-model training, scheduled sampling gradually reduces the probability of restarting ODE integration from training snapshots, ending with trajectories computed from the initial condition.The restart probability begins at 0.99 and is multiplied by 0.99 every 10 epochs.Source: §B.1.4 · pp. 19, 21

B.2 Baseline Implementation

• The appendix describes task-specific baselines, including DeepONet, FNO, and MP-PDE for dynamics modeling, with the baselines implemented autoregressively.For geometry-aware inference, it reports FactorizedFNO comparisons and implementations of GeoFNO, FNO, and UNet.Source: §B.2 · p. 21

C.1 Robustness to Resolution Changes

• CORAL remains competitive and robust on up-sampled Navier–Stokes and Shallow-Water inputs, while MP-PDE overfits the training grid.The tables report up-sampling results for both datasets.Source: §C.1 · p. 22, Table 6 · p. 22, Table 7 · p. 22

C.2 Learning a Dynamics on Different Grids

• Coordinate-based methods generalize better across grids than operator-based and discrete methods; CORAL improves when trained on different grids.DeepONet and MP-PDE perform better or equivalently when trained on one grid, and the authors suggest CORAL may overfit a single shared training grid.Source: §C.2 · p. 22, Table 8 · p. 23

C.3 Training Time

• INR fitting is CORAL’s longest training stage; MP-PDE is the slowest baseline, while DeepONet and FNO are fastest because they require only a forward pass.The comparison reports training times across several datasets.Source: §C.3 · p. 23, Table 9 · p. 23

C.4 Inference Time

• CORAL is inference-time competitive except against operator baselines; MP-PDE slows on denser grids, while DINo is much slower as training and inference grids grow.CORAL’s encoding and decoding are relatively resolution-insensitive and parallel across sensors, while its process stage has fixed dimension; DINo requires 100x more test-time adaptation steps. In these experiments, irregular-grid evaluation required linear interpolation, which slowed inference.Source: §C.4 · pp. 23–24, Figure 6 · p. 24

C.5 Propagation of Errors Through Time

• CORAL’s error grows more slowly than DINo’s during extrapolation, whose error clearly increases out of range.All baselines propagate error autoregressively; MP-PDE and FNO rise quickly early in trajectories, while DeepONet has the highest error except at 100%.Source: §C.5 · p. 24, Figure 7 · p. 25

C.6 Benchmarking INRs for CORAL

• SIREN appears to produce the best dynamics codes among MFN, BACON, and FourierFeatures for Navier–Stokes in-range and out-range prediction.The comparison uses irregular grids sampled from 128 × 128 grids, with training trajectories sharing a grid and test trajectories evaluated on a new grid.Source: §C.6 · p. 24, Table 10 · p. 24

C.7 Impact of 2nd order meta-learning

• 10x lower MSE is achieved by second-order meta-learning, which is more stable than first-order training on Navier–Stokes.The comparison uses the same number of inner steps across three independent runs.Source: §C.7 · p. 26, Figure 8 · p. 26

C.8 Key hyper parameter analysis

• w0 = 30 slightly improves test-set reconstruction.Source: §C.8 · p. 26, Table 11 · p. 26
• dz = 64 yields better in-time forecasting performance.Source: §C.8 · p. 26, Table 11 · p. 26
• Width 256 significantly improves performance across nearly all metrics.The study evaluates reconstruction and forecasting on Navier–Stokes.Source: §C.8 · p. 26, Table 11 · p. 26

D.1 Inverse Design for NACA-airfoil

• Drag 0.04 and lift 0.30 are reached after 50 gradient iterations in the NACA-airfoil inverse-design optimization.The design uses seven spline nodes, maps the input mesh to a pressure field, and minimizes squared drag over lift; optimization yields an asymmetric shape with progressively enhanced lift coefficient.Source: §D.1 · pp. 26–27

E.1 Initial Value Problem

• Figures 10 and 11 visualize CORAL's inferred values on Cylinder and Airfoil.Source: §E.1 · p. 27, Figure 10 · p. 29, Figure 11 · p. 29

E.2 Dynamics modeling

• Figures 12 and 13 show predicted Navier-Stokes and Shallow-Water trajectories across sampling rates, with ground truth in the last row.The predictions span t = 0 to T′; T = 19 and T′ = 39.Source: §E.2 · p. 27, Figure 12 · p. 30, Figure 13 · p. 30

E.3 Geometry-aware inference

• Figures 14–16 visualize CORAL predictions for NACA-Euler, Pipe, and Elasticity.The section also labels pressure on the volume and near the airfoil surface.Source: §E.3 · pp. 27–28, Figure 14 · p. 31, Figure 15 · p. 31
• Figure 9 uses CORAL's inferred airfoil pressure as a surrogate to optimize seven parameters for higher lift and lower drag.The end-to-end procedure uses Adam.Source: Figure 9 · p. 28