Refine Now, Query Fast: A Decoupled Refinement Paradigm for Implicit Neural Fields
Authors: Tianyu Xiong
Skylar Wurster
Han Wei Shen
Venue: ICLR 2026 Poster (2026)
Keywords: implicit neural representation
scene representation network
ensemble simulation
scientific simulation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We propose a "Refine Now, Query Fast" paradigm for INR surrogates, boosting representation fidelity with deep expressive networks at the high inference speed of embedding-based architectures.

Abstract: Implicit Neural Representations (INRs) have emerged as promising surrogates for large 3D scientific simulations due to their ability to continuously model spatial and conditional fields, yet they face a critical fidelity-speed dilemma: deep MLPs suffer from high inference cost, while efficient embedding-based models lack sufficient expressiveness. To resolve this, we propose the Decoupled Representation Refinement (DRR) architectural paradigm. DRR leverages a deep refiner network, alongside non-parametric transformations, in a one-time offline process to encode rich representations into a compact and efficient embedding structure. This approach decouples slow neural networks with high representational capacity from the fast inference path. We introduce DRR-Net, a simple network that validates this paradigm, and a novel data augmentation strategy, Variational Pairs (VP) for improving INRs under complex tasks like high-dimensional surrogate modeling. Experiments on several ensemble simulation datasets demonstrate that our approach achieves state-of-the-art fidelity, while being up to 27$\times$ faster at inference than high-fidelity baselines and remaining competitive with the fastest models. The DRR paradigm offers an effective strategy for building powerful and practical neural field surrogates and INRs in broader applications, with a minimal compromise between speed and quality.

Published as a conference paper at ICLR 2026
REFINE Now, QUERY FAST: A DECOUPLED REFINE- MENT PARADIGM FOR IMPLICIT NEURAL FIELDS
Tianyu Xiong'
Skylar W. Wurster
Department of Computer Science and Engineering
Adobe
Ohio State University
San Francisco, CA, USA
Columbus, OH 43210, USA xi xiong. 3366osu.
swwurstergma com
Han-We Wei Shen
Department of Computer Science and Engineering Ohio State University Columbus, OH 43210, USA shen. 9 94@osu.edu
ABSTRACT
Implicit Neural Representations (INRs) have emerged promising surrogates for large 3D scientific simulations due to their ability to continuously model spatial and conditional fields, yet they face critical fidelity-speed dilemma: deep MLPs suffer from high inference cost, while efficient smbedding-basee models lack suf- ficient expressiveness. To resolve this, we propose the Decoupled Representation Refinement (DRR) architectural paradigm. DRR leverages deep refiner network, alongside non-pan con-parametric transformations, rich representations into compact and efficient embedding structure. This ap- proach decouples slow neural networks with high representational capacity from the fast inference path. We introduce DRR-Net, simple network that validates this paradigm, and a novel data augmentation strategy, Variational Pairs (VP) for improving INRs under complex tasks like high-dimensional surrogate modeling. Experiments on several ensemble simulation datasets demonstrate that our ap- proach achieves state-of-the-art fidelity, while being up 27 faster at inference than high-fidelity baselines and remaining competitive with the fastest models. The DRR paradigm offers an effective strategy for building powerful and prac- tical neural field surrogates and INRs in broader applications, with a minimal compromise between speed and quality.
one-time offline process to encode
INTRODUCTION
Surrogate modeling of large-scale scientific simulations is critical task in fields ranging from cli- mate science to cosmology, enabling accelerated discovery and analysis. Neural Fields, such as Implicit Neural Representations (INRs), have emerged powerful framework for this task, of- fering a continuous and flexible coordinate-based representation. For individual or time-varying fields. INR-based methods have achieved remarkable reconstruction quality and inference speed (Takikawa et al., 2021; Mulllrr et al. 2022) Weiss et al.] 2022) Wurster et al. 2023) Pridovich- Keil et al. 2023 Wurster Shen 2025). However, major challenge remains in scaling these models higher-dimensional ensemble simulations, where collections of outcomes are generated under vary- ing parameters. Existing INR-based surrogates for ensembles struggle to retain both high fidelity and computational efficiency (Li et al. (2025) Chen et al. (2025), presenting challenges to practi- cal applications of INRs for ensemble simulation analysis that balance reconstruction quality and computation efficiency.
This fidelity-speed trade-off is rooted the architectures modern INRs: embedding- based ver- sus MLP-based based approaches. Eibedddig-bbsee methods, which often utilize learnable feature grids,
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achieve high efficiency by encoding information into explicit structure queried via fast interpo- lation (Fridovich-Kaa) et al. 2022 Müller et al. 2022). However, naively scaling these structures to represent both spatial coordinates and simulation parameters incurs prohibitive memory costs. A common workaround employs low-rank or decomposed representations, factorizing the spatial and conditional inputs into separate, low-dimensional structures (Chen et al. [022] Fridovich- Keil iet al. 2023) Chen et al., 2025). While memory-efficien this factorization acts as representational bottleneck, struggling capture complex, non-linear interactions within the data. In contrast, MLP- based architectures can model -igh-ninnniioall functions directly, making them inherently suitable for capturing the intricate relationships in ensemble data. For instance, FA-INR (Li al.] (025)
leverages a Mixture of Experts (MoE) (Jacobs et al., 1991 Shazeer et al. 2017 of MLPs to achieve state-of-the-art fidelity. This expressive power, however, comes at steep computational price: the sequential and heavyweight MIPss, even in MoE setup, result in high inference latency, rendering large-scale data analysis and interactive exploratory workflows infeasible.
Towards resolving the trade-off between representation quality and inference efficiency, we intro- duce a new architectural paradigm: Decoupled Representation Refinement (DRR). The core insight is to enrich INR with deep nonlinear mappings for representation refinement while decoupling the expensive network evaluation from inference. We achieve this with refiner networks and non- parametric transformations that operate directly on the embeddings within the INR's feature struc- tures, rather than on dynamic inputs or intermediate activations, to encode more complex simulation features. Crucially, this refinement process can be precomputed and its results cached, reducing the amortized inference cost to merely that of embedding interpolation and lightweight decoder.
To validate the efficacy of the DRR paradigm, we introduce DRR-Net, simple architecture that im- plements the core principle, and demonstrate that this approach achieves state-of-the-art fidelity and speed against leading INR-based surrogates. Furthermore, training a generalizable INR to model complex field variations across the continuous spatio-conditional space presents a significant chal- lenge, particularly given the prohibitive cost of simulation which results sparse ensemble training data in both the spatial and simulation condition domains. To mitigate this data scarcity, we intro- duce Variational Pairs (VP), a versatile data augmentation strategy that provides robust performance gains across diverse models and datasets.
We propose Decoupled Representation Refinement (DRR), an architectural framework that enhances the expressive capacity of INRs without sacrificing inference efficiency.
We design DRR-Net, a simple architectural implementation that achieves state-of-the-art fidelity and efficiency against INR-based surrogates, validating the efficacy of DRR.
We introduce Variational Pairs (VP), a general-purposs data augmentation for INR datasets that provides significant performance gains for diverse model architectures.
We conduct comprehensive, multi-faceted evaluation, elucidating the fidelity-speed properties of INR models on challenging downstream tasks, including conditional gener- alization and zero-shot spatio-conditional generalization on ensemble simulations, while further demonstrating DRR's versatility in computer vision and graphics applications.
2 RELATED WORK
Implicit Neural Representation (INR) Architectures. INR architectures have evolved from pure MLP-based models more efficient hybrid approaches balance fidelity and speed. While MLPs using Fourier feature encodings (Mildenhall et al.] 20200 Tancik et al. 2020). or sinusoidal and wavelet activations (Sitzmann et al. 22000] Fathony et al. 2020 Saragadam al., 2023 Liu al. 2024), can achieve high fidelity, they remain computationally expensive. We refer to Essakine et al. (2024) for more comprehensive review of state-of-the-art MI.P-based models. Another paradigm instead leverages fast, explicit embedding structures, including dense grids (Martel et al. 2021 Fridovich- Keil et al., 2022. Weiss et al.) 2022 Xiong et al., 2024), hierarchical octrees and hash
grids (Takikawa et al. 202 Müller et al. 2022), and decomposed structures (Chen et al. 2022
Chan et al. 2022. Fridovich Keil et al. [2023] Essakine et al.] 2024). paired with a small decoder
MLP. Our DRR paradigm synthesizes the two philosophies with decoupled refiner to enhance the quality of an efficient embedding structure, representing a step towards more practical INRs that retain both quality and speed deployment.
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INR Acceleration Methods. Orthogonal to the architectural approach for fast INRs, efficiency can also be realized through alternative methodologies. In the context of Neural Radiance Fields (NeRF) (Mildenhall et al. 2020), rendering-speciffe optimizations, such as efficient volume integral computation (Wu et al. 2022) and informed ray sampling (Kurz et al.) 2022 Barron et al. 2022] Gupta al. (023), , are widely studied. However, NeRF-specific algorithms are generally not ap-
plicable to data representation INRs. Conversely, general acceleration techniques such as learning
separate parameters for decomposed domain partitions (Reiser et al. [021 Saragadam et al. 2022
Wurster et al., 2023) or employing knowledge distillation (Reiser et al. 2021 Barron et al. 2022
Duckworth et al. 2024) are highly relevant. Crucially, these strategies are complementary to our ar- chitectural contributions; the DRR paradigm can serve as the efficient backbone model within these decomposed or distilled frameworks. Our work focuses on isolating the fundamental architectural perspective in delivering fast and high-fidelity INR.
Ensemble Simulation Surrogate Modeling. Surrogate modeling accelerates scientific discovery by replacing costly simulations with efficient approximations to explore the parameler-outcomee rela- tionships (Forrester et al.] 2008 Alizadeh et al. 2020). While various machine learning approaches used for tasks like parameter space exploration and sensitivity analysis (Gramacy et al.) [004] Hazarika et al., 2019: Shi et al., (022b]a) a). -NRebased surrogates offer superior flexibility by rep- resenting the ensemble as single continuous function that can be queried arbitrary locations. However, these methods are bound by a clear fidelity-speed trade-off: high-fidelity, MI.P-based models like FA-INR (Li et al. (2025) are computationally expensive, while fast, embedding- -based models like Explorable-INI (Chen et al. 2025) struggle to match the accuracy.
3 METHODOLOGY
3.1 PRELIMINARIES: CONDITIONAL INRS FOR SURROGATE MODELING
Ensemble simulations are critical tools in science, allowing researchers to explore the complex re- lationship between experimental conditions and resulting phenomena, such as yeast cell polariza- tion (Yi et al. 2007 Renardy et al. 2018) or ocean temperature and salinity dynamics (Shi et al.) 2022b). We formally define the resulting dataset as follows. Let P C Rd be the space of input condition parameters. A deterministic simulation, defined by a parameter vector e E P, yields an outcome field t... This field is typically represented as a discrete set of N coordinate-value pairs, 4-= = (Iz) (5))))))) } where each coordinate vector Tj E Rdd and uj E Rd= is the corresponding field value. An ensemble dataset 'D is then a collection of M such fields, generated from discrete set of sampled parameters C = (cI...., CM3 } C P, such that D = {(c, L. )a
INR-based surrogates are a powerful approach for this task (Chen et al. 2025 Li al. 2025 as they learn a single, continuous function, fe. that represents the entire ensemble. This enables flexible data exploration across the continuous domain of conditions e P and coordinates x e Rd This function fo is typically realized using conditional INR architecture, where the spatial coordinate x and condition e are encoded through a spatial encoder (Fxp) and condition encoder (Frond), respectively. Their feature outputs are then fused via a conditioning operation (o) and passed to a final lightweight decoder MLP (g). The entire mapping, with learnable parameters 0 = {0ap: Ocond, } optimized by minimizing the L1 or L2 loss between the network's 's predictions fo(zz, c.) and the ground truth values Uy. can be expressed as:
folæ, c) = (xEsp(x; axp) Econd(e; 1 ccond)s dicc)
3.2 DECOUPLED REPRESENTATION REFINEMENT
central challenge in designing baseb surrogates is balancing reconstruction fidelity and in- ference speed, which are two critical metrics that determine the practical success of any simulation surrogate (Alizadeh et al., 2020). However, current state-of-the-art models fundamentally struggle to resolve this trade-off. The fundamental dilemma in designing INR-based surrogates is rooted in core architectural choice: fast, embedding- based models often act as a representational bottleneck, while expressive, deep MLP-based models suffer from prohibitive inference latency. To resolve this, we propose the Decoupled Representation Refinement (DRR) paradigm, which is founded on the key insight that the expensive computation needed to build high-capacity representation can be
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INR Architecture Performance Implications
DRR INRs (ours)
Feature Grid
Geap Reproseration Rafherar
Chipnal Facio
Stap E: Fascra Gitdl iefarecce
Throughput
Feture
Param-intensive to Scale
Network Expressiveness
Fast Interpolation
Fast Gridi Based Inference
Expressive; Ensy Scaling
Slow Computation
More Expressive
MLP INRs
Per-Param Expressiveness
Figure 1: Our Decoupled Representation Refinement (DRR) paradigm synthesizes the strengths of slow, high-fidelity MLPs and fast, less expressive feature grid, or embedding based models (com- pared similar model sizes) well-rounded design pattern for fast and accurate INRs.
separated from the fast query path required at inference. We begin by detailing the formulation of DRR, followed by analysis of its benefits.
3.2. DRR FORMULATION
Formally, start with baseline embedding-based INR defined by its learnable structure G and query function that produces feature z I(x; 9). The DRR paradigm enhances this three conceptual steps. First, to create richer representation for refinement, we can apply parameter- free transformation to the base structure G, yiclding an intermediate representation, Ç π (G). Second, and most critically, we can introduce a deep, non-linear refiner network, Rg. The refiner's purpose is learn powerful, non-linear transformation of the preprocessed embeddings. We denote this learned transformation as refinement offset, AG:
AG A
(2)
Finally, the complete representation, g', is composed via residual connection. This step sums the learned offset with the preprocessed base, synergistically combining the complementary representa- tions from both:
g' U'= T(g) (=())
(3)
The final feature 2DRR is then extracted from this cached structure using the same efficient query function: ZDRR (xx; G'). Thus, DRR provides an architectural framework for incorporating the expressiveness of a deep neural transformation into computationally efficient embedding structure. During training, all parameters, including the base structure G and the refiner Ry, are jointly opti- mized to minimize the L2 loss for points in fields for training. We now analyze the key benefits that arise from this design.
3.2.2 IMPLICATIONS DRR
The simplicity of the DRR paradigm yields set of architectural properties that resolve the conven- tional sidelity-speed trade-off. We analyze the three primary implications of this design: first, how refinement-dewe -einemeen-ddccoupled inference achieves the speed embedding bbseed models; second, how the framework promotes synergistic representation learning to improve expressiveness; and finally, how the preprocessing stage allows for further representation enhancement.
Deeinement-Decoupple Inference. The foremost advantage of the DRR paradigm is its ability to deliver the expressivity of a deep network while matching the inference efficiency purely embedding- basedd models. During inference, the deep refiner network R.g is evaluated only once
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in pre-computation step produce and cache the static, refined structure G' with Eq. After this one-time cost is paid, the refiner is discarded. Consequently, all subsequent queries operate solely on the cached structure, reducing the amortized computational cost of inference to that of a standard emhedding- based INR: a simple interpolation and a lightweight decoder.
Synergistic Representation Learning. DRR enhances expressivity by resolving the "entangled representation" problem. In standard INRs, embeddings must simultaneously capture global struc- ture and local detail. The refiner decouples this by learning a shared decoding function across the grid, freeing the base embeddings to specialize in encoding dense, latent high-frequency signals. This creates tightly coupled system: the base acts as a specialized pre-conditioner, optimizing for the refiner to suppress aliasing and decode physical quantities, rather than functioning as a stan- dalone approximation (see Sec.
Second, the refiner acts an offline fusion module. While common embedding- based INRs burden the lightweight MLP decoder with fusing multi-scale features (Müller et al.| /Weiss et al. [022] Wurster et al. 2023), DRR offloads this complexity by learning to fuse features offline. By baking these refined representations directly into the cached grid G', we allow the online decoder to remain lightweight while operating rich, pre-fused features.
Embedding Preprocessing with Non-Parametric Refinements. To ensure the proper optimiza- tion of the deep refiner network, the richness of the input information and the quantity of data points are critical factors. Since the refiner operates directly on the feature structures, these require- ments correspond to the embedding dimensionality and the grid resolution, respectively. Satisfying these needs in a standard setup would necessitate base grids with high resolutions and large em- bedding dimensions, but such design introduces a prohibitive parameter overhead. To resolve this trade-off and facilitate refiner optimization, we apply non-parametric transformations to synthesize a high-capacity feature manifold without increasing storage costs. We leverage two strategies. First, Structural Super- Reseurtinotti deterministically upsamples the grid to increase feature density. Sec- ond, feature upsampling with Positional Encoding (Mildenhall al., 2020)) projects features into a higher-dimensions space to capture complex frequencies. As validated in Sec, [F] these operations serve as critical enablers that allow the refiner to generalize effectively while maintaining a compact base structure.
3.3 INSTANTIATING THE DRR PARADIGM WITH DRR-NET
To validate our paradigm, we introduce DRR-Net, a concrete and straightforward instantiation, as illustrated in Fig. 2 designed to demonstrate how the general DRR formulation be adapted to create a powerful surrogate for ensemble simulations.
The Hulti-Resolution Unification Principle. Modern, nigh-peeformance embedding structures frequently rely on multi-resolutionn representations (Müller et al. 2022, Wurster et al. 2023). To apply DRR to such structures, we introduce general unification principle. The core challenge is preparing a single, unified input for the refiner from a set of embeddings {g' H/= at different resolutions. Our solution is a preprocessing function, Tunify, that first upsamples each structure g' to a common resolution via an operator Ut', the structural super-resolution step, and then concatenates their features channel-wis -isse (():
L 1=1
Gunitied = Tunify Tunity (19))
(g')))
(4)
This unification is a crucial mechanism that allows a deep refiner to learn from and enhance the complex, multi-scale representations from the modern embedding structures.
Application to DRR-Net Encoders. The DRR-Net architecture, shown in Fig. 2 applies this unifi- cation principle to two distinct branches: a spatial encoder and condition encoder.
The Spatial Encoder, depicted in Fig. bb) begins with set of Teup multi-resolution feature grids, 12r. To explicit, each of the Lap grids has its own spatial resolution but shares a common feature dimension of djs at each vertex. As the first step of unification, each grid G is upsampled to common, high resolution via structural super-resolution using interpolation. Then, as shown in E upsampled features from all Lup grids are concatenated channel- at each vertex. This pro- cess produces single, dense, unified grid, Sunified with vertex embeddings of dimension I. x dys.
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Learnable Composent
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Figure 2: Concrete instantiation of the DRR paradigm via DRR-Net. Part (a) illustrates the overall architectural components, while part (b) and (c) detail the unification and refinement procedure applied to the base multi-resolution embedding structures.
Optionally, P.E. feature upsampling (with Kee frequencies) can then be applied to this unified repre- sentation, lifting its internal dimension to Lep djs x 2K The resulting grid serves as the input to the spatial refiner, which produces the final refined grid S'pp To encode an input coordinate æ, is interpolated from this refined grid G2p. producing a single embedding of the final dimensionality.
The Condition Encoder, depieted in Fig. [ee), is designed be scalable to arbitrary number of simulation parameters. For a simulation with de parameters, the encoder initializes de sets of Leand multi-resolution ID feature lines, ( each with a base feature dimension of d These structures are unified via hierarchical, "local-then-goobl" process to learn crucial cross-parameter feature fusion lacking in the low-rank setup. First, an parameer (local) unification step is applied to the Leand lines for each parameter separately. It upsamples each parameter's 's lines their own common (highest) resolution and concatenates their features, producing d, independent, unified feature lines, (% ) k== Each of these lines now has feature dimension of T.cond x d Second, an inter-paramett (global) unification step unifies these de lines across all parameters. As shown in 5 this step upsamples all de lines to single, global high resolution (via u+) and concatenates them channel-wise.
Geond = Tanify (Hg 1d: 1 = [| 11 (gt
(5)
This results in the final, consolidated ID representation, Geonds which is then processed by the condition refiner. Crucially, after refinement, this single line is split back into de dedicated feature lines, one for each parameter, which now contain richly fused cross-resolution and tross-paramett features. To encode the de-dimensional input parameters, we interpolate from each of the refined lines, yielding de feature vectors. Since the refiner's output dimension matches its input as with the residual formulation, each vector has a size of Leond X dfc- These are then concatenated to produce the final conditional feature for the decoder, with a total dimension of de X Lcand X dfc.
3.4 DATA AUGMENTATION WITH VARIATIONAL PAIRS
Training INR surrogates on a sparse set of ensemble members is challenging, making data aug- mentation a crucial tool for improving generalization. A related technique, Variational Coordinates (VC) (Zhang et al. 2024), was introduced to perturb input coordinates for anti-aliasing. However, we find that naively applying VC as augmentation strategy can be ineffective or even harm model accuracy. We hypothesize this failure stems from a substantial mismatch between the generated data and the ground truth. Specifically, while the true function that maps coordinates to values in a sci- entific simulation is often highly complex and continuous, the VC augmentation implicitly enforces piccewise-constant assumption by keeping values unchanged for perturbed coordinates. Our core

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insight is to resolve this mismatch. We propose family of augmentations, Variational Pairs (VP), that generates a new, physically-plausiblee value corresponding to the perturbed coordinate.
VP-S: Spatial Augmentation. Our first strategy, VP-S, is general-purpose augmentation applica- ble to any INR. It generates a perturbed coordinate ī = z tz, where the perturbation eg is sampled from zero-mean isotropic Gaussian distribution, c, (N(0, and is truncated at distance threshold r. This follows the formulation of VC (Zhang et al. [202)). Crucially, we approximate the new value by interpolating the original field &e this new location. This interpolation function, I[ T) is a weighted average of the ground truth values from a local set of neighboring vertices, (z)), on the original simulation grid:
Σ (1(())
JEV(2)
where the set of neighbors Vo) and their corresponding weights wy (z) are determined by the cho- sen interpolation function. While any interpolation imposes its own prior, this assumption of local smoothness is much milder and more physically plausible than the piecewise-constant one. The resulting pair (ž, ū) therefore provides a higher-quality supervisory signal for learning a locally smooth and better representation of the field than VC.
VP-SC: Spatio-Conditional Augmentation. For the specific challenge of conditional INR tasks like ensemble surrogates, we propose a more integrated strategy, VP-SC, that augments both spatial and conditional inputs. Given a sample from a field &e.. we generate a perturbed query (ī, c) by adding truncated Gaussian noise both the coordinate and the condition parameter. The key challenge is to approximate the unknown value i for this novel condition č. We estimate it via two-stage interpolation scheme. First, we identify the K-nearest neighbor conditions {cx 1 to the original condition e; from the training set C. We perform a spatial interpolation for the perturbed coordinate I within each of these K neighbor fields using Eq. (6] yiclding a set of candidate values {v/ = ( [i) I== Second, we perform a conditional interpolation these candidate values using Inverse Distance Weighting (IDW) to compute the final value:
K
1/llē cx112
wR (o)uk
where wk (2)
E/=1 1/llē- cyl12 cjll2
k=1
This process is designed to generate plausible training samples in the continuous spatio-conditional space, providing the model with a more diverse data distribution than the original sparse samples. The goal is to encourage the model to learn the underlying ensemble dynamics more effectively and improve its generalization to unseen conditions.
EXPERIMENTS
We conduct series of experiments to rigorously evaluate our proposed DRR paradigm and DRR- Net architecture. First, we benchmark DRR-Net against state-of-the-art baselines on the primary task of conditional generalization, assessing both fidelity and efficiency (Sec. 4.2 with additional vi- sualizations in Sec. S). Second, we introduce more challenging test of zero-shot spatio-conditional generalization to probe the quality of the learned continuous representation (Sec. (43). Finally, we present an ablation study over the data augmentation strategies to evaluate our proposed VP methods (Sec. 4.4 more comprehensive suite of DRR components, including the refiners Rgg, preprocess- ing functions and VP hyperparameters are provided in the appendix (Sec. Eland beyond).
To demonstrate the versatility of the DRR paradigm beyond scientific surrogates, we also present its application to broader vision and graphics tasks in Sec. [
4.1 EXPERIMENTAL SETUP
Datasets. We evaluate our method on three 3D ensemble simulation datasets, consistent with prior work in INR- surrogate modeling (Chen et al. 2025 Li et al. 2025). These include the cosmo- logical simulation Nyx (Almgren et al., 2013), the oceanography simulation MPAS MPAAS-Ocean (Ringler et al. 2013), and the hydrodynamics simulation Cloverleaf3D (Mallinson et al. 2013). Key statis- ties for each dataset, including the number of ensemble members, parameter dimensions, and spatial resolution, are summarized in Tab. with more details in Sec. B.4
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Table I: Description of ensemble simulation datasets used for evaluations.
Parameters
Resolution
Training
Testing
Nyx
3D Cariesian Grid
Durk Matler Density
3
256
100
30
MPAS-Ocean
Voronoi Mesh
Ocean Temperature
4
11841166 vertices
70
30
Cloverleaf3D
Cartesian Grid
Energy
128
500
Baselines. We compare DRR-Net against three state-of-the-art methods designed for high- dimensional fields and ensemble simulations, which represent the main INR architectural ap- proaches. For efficient, embedding- baased models, we compare against (1) K-Planes (Fridovich-Kell al. 2023), which uses decomposed 2D feature planes, and (2) Explorable-INR (Chen al. 2005)) which uses a factorized grid representation. To represent high-fidelity, MLP-basee models, we com- pare our approach against (3) FA-INR (Li et al. 2025). concurrent work that leverages a mixture of MLP experts for spatial feature encoding with pooss-attention--o based conditioning.
Metrics. We evaluate all models on both computational efficiency and prediction fidelity to as- sess their practicality as simulation surrogates. For efficiency, measure four key indicators: the computational cost of inference (TFL.Oss per 10 points) assess theoretical complexity; the total inference time (in seconds) to simulate data analysis latency; the total training time (in hours); and the model size (number of parameters). For fidelity, we report the Relative L2 error (Rel L2), Peak Signal-to-Noise Ratio (PSNR), and the Structural Similarity Index Measure (SSIM) (Wang et al. 2004). As the standard SSIM computation relies on a sliding window over a regular grid, we do not report it for the MPAS-Ocenn dataset with an unstructured mesh topology.
Table 2: Results for the conditional generalization task. DRR-Net achieves a state-of-the-arr fidelity and efficiency all datasets, with more pronounced benefits for Nyx and Cloverleaf3D.
TFLOPs 108 pts 4
Inference Time (sec)
Training Time (hr)
Dataset
Model
Rel 1.2,
PSNRT
SSIM
Params
K-Planes
1.96-01
28.86
0.797
57.0
21.6s
23,8h
12.1M
Nyx
FA-INR
3,95e-02
42.79
0.975
2569.2
287.2s
64.
9.5M
Explorable- INR
4.64e-02
41.39
0.972
39.6
9.6s
24.7h
14.7M
DRR-Net Net (ours)
3.18e-02
44.69
0.986
57.2
10.7s
24.0h
8.9M
Planes
3.81e-01
16.9%
6.6h
13.5M
MPAS-Ocean
FA-INR
5.58e-03
52.13
422.1
89.8s
21.1h
1.M
Explorable-INR
1.13e-02
46.03
39.6
5.3x
3.3h
14.7M
DRR-Net (ours)
7.76e-03
49.26
47.3
4.8s
7.2h
4.6M
K-Planes
8.42c-01
30.02
0.787
67.7
17.7s
26.7h
6.8M
FA-INI INR
1.11e-01
47.60
0.991
422.1
56.2s
29.4h
L.M
Cloverleaf3D
Explorable-INR
1.46c-01
45.22
0.984
39.6
5.3s
28.0h
4.5M
DRR Net (ours)
9.81e-42
48.69
0.994
52.6
5.58
44.0h
0.9M
4.2 CONDITIONAL GENERALIZATION EVALUATION
Our first experiment evaluates DRR-Net on the primary task of conditional generalization, assess- ing its ability predict entire fields for unseen simulation parameters. The results, presented in Tab. 22 demonstrate that DRR-Net successfully overcomes the conventional trade-off between fi- delity and efficiency. On both the Nyx and Cloverleaf3D datasets, DRR-Net achieves state-of-the-an reconstruction quality while maintaining inference speeds competitive with the fastest embedding- based methods. For example, on Nyx. DRR-Net obtains the highest PSNR (44.69), yet is over 27 x faster during inference than the high-fideltyy FA-INR Furthermore, DRR-Net is highly parameter- baselines.
efficient, achieving these results with a smaller model size than all other embedding-
On the MPAS-Ocean dataset, we observe that the MLP-based FA-INR achieves a higher PSNR. We attribute this to a structural mismatch between the dataset's underlying geometry and the inductive bias of grid-based methods. MPAS-Ocean is simulated on unstructured spherical mesh with an adaptive vertex density, whereas feature grids, as used in our and other embedding based models, in- terpolate in Cartesian system assuming uniform grid spacing. An MLP, lacking a spatial inductive bias, is less affected by this mismatch. Crucially, however, DRR-Net still significantly outperforms the other grid-based method, Explorable-INR, by 3 dB in PSNR. This provides strong evidence that
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our DRR paradigm is effectively compensating for the inherent limitations of the base grid repre- sentation, demonstrating its robustness. Additionally, we demonstrate through visualization that our DRR-Net can qualitatively capture temperature features well FA-INR in Sec. C.3 while being substantially faster.
Table 3: Spatio-conditional generalization results. DRR-Net consistently attains the best fidelity
with highly competitive inference efficiency matching the embedding-on model Explorable-
Unscen Fields
Trained Fields
Inference Time (sec)
Dutaset
Model
# Params
(Rel PSNR1 (SSM)))
(Rel1.2L/PSNEF SSIM))
K- K-Planes
1.89c-01 29.18/ 0.709
2.91e-01 / 29.62 70.7
75.3s
3.9M
FA-INR
4.55e-02 41.57 00971
7.29c-02/ / 41.65 / 00971
1245,1s
9.5M
Nyx
Explorablc-INR
6.48c-02/ / 38. 50/ 0.959
8,44c-02 / 40.38 / 0.963
38.65
10.0M
DRR-Net (ours)
4.31e-02/ 42.04 0,975
7.07e-02/ / 41.92/ / 0.975
43.7%
8.7M
K- K-Planes
4.47c-01 / 35.52 0.914
5,78c-01 / 36,04 / 0.919
81.5s
3.1M
Cloverleaf3D
FA-INR
1.07c-01 47.99/ 00.922
9.15c-02/ / 52.06 / 0.996
327.8s
1.M
Explorable-INR
1.25e-01 46.55/ 0,989
1.10c-01 50.49 / 00.994
27.3s
4.0M
DRR-Net (ours)
1.01e-01 /8.43/ 0.993
8.70c-02 52.49 / 0.996
25.65
0.9M
4.3 SPATIO-CONDITIONAL GENERALIZATION EVALUATION
Having established DRR-Net's performance on the standard task, we now probe the fundamental quality of INRs with more challenging test of spatio-conditional generalization. To do this, evaluate all models on zero-shot super-resolution for both trained and unseen ensemble members. Models are trained on fields that have been downsampled to x2 lower resolutions and are then evaluated on their ability to reconstruct the original, full-resolution fields. For example, Nyx training fields are downsampled to 128 and evaluated 256 This task is crucial assesses the model's ability to learn truly continuous representation across both spatial and conditional domains. We also evaluate the condition-one only generalization directly on the downsampled resolution in Sec. too study the fidelity and efficiency of models in smaller-scale problems.
The results, presented in Tab. 3. demonstrate that DRR-Net's superior performance holds. For both unseen and trained fields, we report the relative L2 error, PSNR, and SSIM in single cell. On both the Nyx and Cloverleaf3D datasets, DRR-Net achieves the highest reconstruction quality for both trained and test simulation parameter settings at the unseen full resolution, surpassing the highest- fidelity baseline FA-INR while being 11-28x faster to generate the fields. It shows that the bene- fits of our DRR paradigm generalize robustly across both spatial scales and condition parameters, demonstrating the efficacy of DRR-Net as a continuous implicit representation.
4.4 ABLATION STUDY: DATA AUGMENTATION
We conduct ablation study to validate our proposed Variational Pairs (VP) data augmentation. We compare our two variants, VP-S and VP-SC, against a no-augmentation bascline and the Variational Coordinate (VC) method (Zhang al. 2024). The analysis. presented in Tab, 4 is performed across all baseline models on our two primary Cartesian grid simulations, Nyx and Cloverleaf3D.
First, the results confirm the risks of the naive VC approach. Its performance is highly unpredictable; while it sometimes offers marginal benefit, it is detrimental to DRR-Net on both datasets, and this effect more pronounced for Explorable-INR in Cloverleaf3D. This supports our reasoning that the implicit piecewise-constan assumption provides low-quality supervisory signal.
In contrast, our proposed VP methods provide consistent and substantial performance gains across the vast majority of -dtel-dataset combinations. The simpler VP-S strategy emerges remark- ably robust and effective all-arounder. On Cloverleaf3D. VP-S the top-performing augmentation for three of the four models, including our own DRR-Net (+1.79 dB). The more complex VP-SC also proves effective, delivering the best performance for the baselines on Nyx, though its additional complexity does not always translate to superior gains over VP-S. This suggests trade-off between the methods. We note that the efficacy of VP methods can be further influenced by hyperparameter tuning, detail in Appendix H

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Table Ablation study of data augmentation strategies, showing that proposed VP methods consis- tently outperform the no-augmentation and VC baselines acToss diverse INR architectures.
Model
Augment Method
Nyx
Cloverleaf3D
Rel 1.24
PSNRT
SSIM
R.2
PSNR1
SSIM
None
2.07c-01
28.41
0.783
8.57c-01
29.87
0.784
K-Planes
VC
1,98c-01
28.80
0.792
8.87c-01
29.57
0.775
VP-S (ours)
2.05e-01
28.48
0.804
7.89e-01
30,59
0.801
VP-SC (ours)
1.96e-01
28.86
0.797
8.42e-01
30.02
0.787
Nome
4,96e-02
40.81
0.961
6.40e-01
32.40
0.764
FA-INR
VC
4.91c-02
40.91
0.963
1.0c-01
48.47
0.992
VP-S (ours)
4.31e-02
42.04
0.970
9.33e-02
49.13
0.993
VP-SC (ours)
3.95e-02
42.79
0.975
1.11e-01
47.60
0.991
None
8.14e-02
36.51
0.932
1. 1.50c-01
45.03
0.984
VC
5.71e-02
39.59
0.964
2.02c-01
42.41
0.980
Explorable-INe
VP-S (ours)
4.69c-02
41,30
0.971
1.67c-01
44.07
0.978
VP-SC (ours)
4.64e-02
41.39
0.972
1.46e-01
45.22
0.984
None
3.09c-02
44.92
0,986
L.3c-01
47.47
0.993
DRR-Net (ours)
VC
3.81e-02
43.11
0.982
1.15c-01
47.29
0.992
VP-S (ours)
3.20c-02
44.61
0.985
9.19e-02
49.26
0.994
VP-SC (ours)
3.18c-02
44.69
0,986
9.81c-02
48.69
0.994
5 LIMITATIONS AND FUTURE WORK
While our work demonstrates the effectiveness of the DRR paradigm, we identify several limitations and promising avenues for future research that build upon our findings.
Application to Embedding Structure Variants. Our implementation of DRR-Net is limited to multi-resolution Cartesian feature grids. However, the DRR paradigm itself is agnostic to the spe-
cific embedding structure. A compelling direction for future work is to apply DRR to other advanced representations, such as hash grids (Müller et al., (0022), adaptive grids (Wurster et al. 2023 for sci- entific simulation, or other learned primitives like 3DGS (Kerbl et al. 2023).
Alternative Refiner Architectures. In this work, we only used simple, point-wise GLU-based refiner to isolate the benefits of the DRR framework and establish a baseline. significant oppor- tunity exists in exploring more specialized refiner architectures. For instance, CNN-based refiner could explicitly leverage the spatial locality of the embedding grids, while a Transformer- based re- finer could model global, long-range dependencies between embedding features. Such architectures, guided by the DRR principle, could potentially unlock further performance improvements.
Scope of Generalization. Current INRebased ensemble surrogates, including this work, primarily target conditional generalization within the convex hull of the training parameters or interpolation. This scope aligns with standard surrogate modeling workflows where the parameter space of interest is pre-defined for exploration. However, extending INR surrogates to perform robust extrapolation, or predicting simulation behaviors outside the training range, presents distinct challenge, particu- larly for data-driven approaches in modeling such complex non-linear dynamics. We identify this as promising direction for future research to further enhance the utility and generality of neural field surrogates.
6 CONCLUSION
In this work, we addressed the critical trade-off between fidelity and efficiency in INR-based sur- rogates for large-scale ensemble simulations. We introduced the Decoupled Representation Re- finement (DRR) paradigm for INR, a novel framework that leverages deep refiner network in an offline stage to enhance the expressive power of a compact and efficient embedding structure. This approach successfully decouples the high representational capacity from the fast inference path. Our comprehensive experiments demonstrate that our implementation, DRR-Net, achieves state-of-the- art reconstruction quality on several scientific benchmarks while remaining efficient, similar to the fastest embedding- based methods. We believe that the DRR paradigm offers general and effec- tive strategy towards building the next generation of powerful and practical implicit neural field surrogates, paving the way for more efficient scientific data analysis and discovery.
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ACKNOWLEDGMENTS
We gratefully acknowledge the financial support provided by the U.S. Department of Energy (DOE) SciDAC program under grants DE-SC0021300 and DE-SC0023193. Additionally, this research was supported by the National Science Foundation (NSF) Division of Information and Intelligent Systems (Grant IIS- (S5-1957764 and by Los Alamos National Laboratory under Contract C3435.
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Linear
Features
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Input
GLU Blocks
Linear
Linear
ReLU
Refiner Architecture
Figure Architecture of the refiner network, which consists of stacked GLU blocks with pre- normalization and ReLU intermediate activation.
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A LLM USAGE ACKNOWLEDGEMEN
We acknowledge using large language model (LLM) to improve the final manuscript by correcting grammar, addressing typographical errors, and consolidating phrasing for conciseness and clarity. The authors take full responsibility for all content.
B IMPLEMENTATION DETAILS
Our source code is publicly available at http::://gittaab..omo DDR This section provides further implementation details for the experiments presented in Sec. 4
B. REFINER GLU ARCHITECTURE
As described in Sec. 3.3 both refiners from the spatial and condition branches are composed of GLUs. We illustrate the detailed architecture in Fig. 3 Within each GLU block, the input is first processed through pre-normalization layer using Root Mean Square Normalization (RMS Norm) //hang & Sennrich] 2019). The normalized output is then branched into two parallel paths con- path ReGLU formulation (Saazeer 2020). The two sets of features are then multiplied, followed by a residual connection from the unnormalized input.
taining separate linear transformations. and ReLU is applied
This ReGLU representative modern architecture for point-wise transformation, and the addi- tional normalization and residual connections ensures both the refiner and the embeddings ceive proper level of gradients, which can be critical since they are among the earliest components of DRR network, prone vanishing gradient problems.
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B.2 SOFTWARE AND HARDWARE
All of models and the baselines are implemented in Python using PyTorch. Crucially, tomized computing kernels or hardware-specific optimizations were used for any model. This en- sures that all performance comparisons are based on architectural and algorithmic efficiency rather than low-level code optimization.
All experiments were conducted on a single node of high-performance computing (HPC) cluster. Each experimental run in Sec. was allocated 10 CPU cores (AMD EPYC 7H12), 128 GB of system RAM, and single NVIDIA A100 GPU with 40GB of VRAM.
B.3 EVALUATION METRICS
Because computational efficiency is one of the primary motivations for building surrogates (Al- izadeh et al., 2020), we reported suite of metrics over Sec. This section provides the detailed procedures used to calculate these efficiency metrics.
FLOPs. We report the theoretical computational cost in TFLOPs (trillions of floating-point oper- ations) per 10 queried points. We chose this unit to approximate the cost of typical large-scale analysis tasks, such as reconstructing a single high-resolution 1024 field or exploring a parameter space of 500 members 128 resolution.
To obtain a stable value, we first compute the average FL.OPs for a forward pass on a representative batch of 102 conditioning parameters and 10 coordinates over 101 runs. We then compute stable average and scale by 10 to get the final reported number for 10 points.
Inference Time.
We report the wall-clock time required reconstruct the full test set for each task. For the conditional generalization task (Tab, ( this is the time to evaluate all unseen ensemble members. For the spatio-conditional task (Tab. 37. this is the time generate the full-resolution fields for all members, since none were seen at high resolution during training.
Training Time. Measuring the exact end-to-end training duration can be inconsistent on HPC systems due to job scheduling. To ensure a fair comparison, we report stable lower-bound estimate: we measure the average per-iteration time over 100 steps (once training is stable) and multiply by the total number of training iterations. This serves a consistent lower-bound estimate of the training time, excluding initializations, data-loading, and other overheads.
B.4 DETAILS OF SIMULATION PARAMETERS FOR DATASETS
In this section, provide detailed description of the three 3D ensemble simulation datasets evalu- ated in this work. For all datasets, each ensemble member represents a transient simulation evolved over specific time duration, with the scalar field of interest extracted from the final timestep. Across the ensemble, all simulation conditions remain constant except for the specific parameters selected for exploration. We detail the dataset-specifie configurations below. For train/test splits and statistical overviews, please refer to Tab. [ in the main text. Additionally, we provide the exact pa- rameter value combinations for all training and testing members in the dataset /epaaaaaaa. marrmmssuuce directory of the open-sourced code.
Nyx (Almgren al. 2013) is a cosmological hydrodynamics simulation developed by Lawrence Berkeley National Laboratory. The ensemble varies three cosmological parameters of interest: the total matter density (OmM E [0.12, 0.155)), the baryon density (OmB E [0.0215, 0.0235)), and the Hubble Constant (h E |0.55, 0.85). A total of 130 ensemble members were generated via random sampling within these parameter ranges. Each simulation was evolved for 200 timesteps.
MPAS-Ocean (Ringler et al. 2013) is an unstructured-mestt ocean model developed by Los Alamos National Laboratory. The ensemble explores the sensitivity to four parameters: the Bulk wind stress Amplification (BwxA |0.0, 5.0).. the Critical bulk Richardson number (ChrN E 10.25, 1.00)). the Gent-MeWilliams Mesoscale eddy transport parameterization (GM |600.0, 1500.0)), and the horizontal viscosity (HV [100.0. 300.0).. The simulation utilizes spherical coordinate system with a Voronoi tessellation for the horizontal grids at each depth level. The dataset consists of 100 members, each simulated for duration of 15 model days.
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Table 5: Architectural hyperparameters for INR surrogate models across the evaluated ensemble simulation datasets.
Model
Hyperparameter
Nyx
MPAS-Ocean
Cloverleaf3D
Spatial Plane Base Resolation
32
32
16
Spatial Plane Multi-Re Res Multipliers
[,,2,488
2,,2,488
2,,2,488
Condition-Onn Plane Resolution
25
25
25
K-Planes
Plane Embedding Dim
32
32
32
Decoder MLP Hidden Dim
128
128
128
Decoder MLP Layers
3 layers
layers
3 layers
Total Parameters
12.1M
13.5M
6.8M
Spatial Gate Grid Resolution
Spatial Gate Grid Emb. Dim
Condition Feature Line Resolution
Condition Line Emb. Dim
64
Num Experts
Expert Encoder MLP Layers
layers
layers 256
layers
FA-INR
Expert Encoder MLP Dims
512
256
Expert Num Embeddings
1024
Expert KV Embeddings Dim
Expert Top
Decoder MLP Hidden Dim
128
128
Decoder MLP Layers
3 layers
layers 1.0M
layers
Total Parameters
9.5M
1.M
Spatial Feature Grid Resolution
64
64
48
Spatial Grid Embedding Dim df.
32
32
32
Spatial Feature Plane Resolution
256
256
96
Spatial Plane Embedding Dim
32
32
32
Explorable-INR
Condition Feature Line Resolution
16
16
16
Condition Line Embedding Dim
16
16
16
Decoder MLP Hidden Dim
128
128
128
Decoder MLP Layers
3 layers
layers
layers
Total Parameters
14.7M
14.7M
4.5M
Spatial Feature Grid Multi-Res
112, 128]
[64, 128]
[16, 48]
Spatial Grid Embedding Dim
2
2
Spatial SSR Max Grid Res
128
Spatial Num Freqs Kp
Spatial Refiner Hidden Multiplier
Spatial Refiner Layers
3 layers 8,
layers
layers
DRR-NC Net
Condition Feature Line Multi-Re
[4, 8, 16, 32
[2, 8, 16]
(Ours)
Condition Line Embedding Dim dye
Condition Refiner Hidden Multiplier
5.34 4 layers 128 layers 8.9M
2.67
Condition Refiner Layers
layers 128
layers
Decoder MLP Hidden Dim
128
Decoder MLP Layers
layers 4.6M
layers 0.9M
Total Parameters
CloverLeaf3D (Mallinson et al., 2013) Lagrangian-Fulerinn hydrodynamics simulation. The ensemble varies six parameters governing the initial state: three density values (Density 1-3) and three energy values (Energy 1-3). The parameter sampling is subject to a hierarchical constraint: the values are ordered such that Energyi < Energyz Energys and Density: Densityz < Densitys- Furthermore, a cross-variae varialvariable constraint enforces that Density; < Energy: for each corresponding component
f,,2,33 3). The dataset comprises 600 ensemble members, with each simulation evolved for 200 timesteps.
B.5 MODEL. IMPLEMENTATION AND TRAINING PROCEDURES
Model Hyperparameters. For all baselines, we utilize their official implementations. We adhere strictly to the recommended hyperparameter settings provided the original works for the evaluated datasets, such Explorable-INe -INioraaabe-RRRRRR and FA-INR (Chen et al. 2025 Li al. 2025). As K-Planes was not originally benchmarked on these specific datasets, we adapted its configuration to ensure its feature plane resolutions match the data resolution. We enumerate the detailed model configurations
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Table 6: Training hyperparameters for each dataset.
condition batch size No
coordinate batch size N₂z
condition- impsmp
coordinate-
Dataset
epochs
impsmp
Nyx
50
216
MPAS-Ocean
50
16
2
Cloverleaf3D
50
32
212
X
in Tab. [5] Full reproducibility details are also provided in the open-sourced experiment configuration files within the configs directory. For deeper architectural specifics of the baselines, we refer readers
to the original publications (Chen et al. 2025 Li et al. 2025)-
Regarding the DRR-Net specifications in Tab. the Multi-Res entries denote the resolution hier- archy of the base structures. For instance, [64, 128) indicates two distinct 3D feature grids with resolutions of 641 and 128 respectively. The input dimension to the refiner is determined by the concatenation of these embeddings following the r-transformations described in Sec. 3.3 Taking the Cloverleaf3D spatial encoder configuration as a concrete example, with 4 grids each embed- ding size 2 and P.E. frequeney of 6, the final unified embedding size yields x x (2 x 6) 96. As a GLU-based architecture, the refiner projects these features to a hidden width defined by the listed multiplier. In this case, a multiplier of approximately 2.67 expands the 96-dimensional input hidden bottleneck dimension of 256. We select these non-integer multipliers specifically to align the hidden dimensions with powers of for computational efficiency.
Training Procedure and Hyperparameters. To ensure a fair comparison, all models, including DRR-Net, trained with the Adam optimizer (Kingma & Ba 2014) using base learning rate of 1 x 10 All models use a cosine annealing scheduler (Loshchilov & Hutter (016) to decay the learning rate by two orders of magnitude over the course of training. Following the official implementation, the MLP-based spatial feature encoders in FA-INR use a separate, smaller learning rate of 1 x 10-5 5 for the Nyx dataset. Each training batch consists of coordinates sampled from No fields, with N₂ coordinates per field. Each INR surrogate model is trained to minimize the L2 loss for the ground truth scalar value of scientific variable for the given coordinate and simulation parameter input. We present comprehensive summary of all hyperparameters used for training every model on all datasets in Tab. 6
For the detailed discussion on the hyperparameter settings for our data augmentation methods VP-S and VP-SC, we refer to the subsequent ablation study section in Sec. H
Data Preprocessing and Normalization. Prior to training, normalize all data inputs and out- puts. Input spatial coordinates (x) are scaled to the range [-1, 1]dd to be compatible with standard PyTorch grid sampling functions. Condition parameters (c) min-max normalized to [0, 1]ss the normalization range is determined by the global minimum and maximum values across both the training and testing sets to define the full range of interest for parameter space exploration. Finally, the ground truth field values (v) are also min-max normalized, where the statistics (min and max) are computed solely from the training set data following standard practice. We note one dataset-specific step: for the Nyx dataset, logarithmic transform is applied to the Dark Matter Density values to better handle its large dynamic range.
Driveen Data Sampling. For our main experiments, we follow the importance-drivee sampling strategy of prior work (Chen et al.) 2025). As reasoned by [Cen al. (2025). the technique could improve training by focusing on more complex or less frequent regions of the data. The importance score based on the inverse frequency of values (Biswas al. 2018 2023). For each ensemble member, histogram of its field values is constructed, and each coordinate's importance is defined by the inverse frequency of its value. The importance of an entire member field is the sum of its coordinate scores, normalized across all members. In order to keep experiment settings consistent with Explorable-INR and FA-INR, for the conditional generalization task, we also use these scores to preferentially sample both the ensemble members and the coordinates within them.
Uniform Random Sampling for the Super-Resolution Task. For the novel spatio-conditional generalization task presented in Sec. 4.3 we do not apply importance sampling. As this is new
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Ground Truth
DRR-Net (Ours)
Explorable-ING
FA-INR
Visualization Fidelity:
PSNR1: 27.85
SSIN 0.9921
PSNRT: 21.59
SSIM 0.9245
PSNR1: 24.30
SSIM 0.924
Model Efficiencyi
Timel: 0.00s
Params, 0.65M
Timel: 0.04s
Params].: 4.40M
Time.: 0.54s
Params]: 0.97M
Visualization Fidelity:
PSNR": 31.04 SSIM
PSNRT: 22.68
SSIM 0,5360
PSNRT:
SSIM
Model Efficiency:
Timel:
Time::
Params
Time::
Params]:
Figure 4: Visual comparison dataset. Reconstructed 1283 points in just 0.05 seconds, our DRR-Net better captures complex high- energy and high-density structures in respective datasets than competing state-of the-art surrogates, demonstrating superior balance of fidelity and speed.
unseen field from the Cloverleaf3D (top) and Nyx (bottom)
evaluation task, we opt for a simpler, strong bascline using uniform random sampling across both the conditional and spatial domains.
C DATA VISUALIZATIONS FOR QUALITATIVE EVALUATION
This section provides additional qualitative comparisons for the conditional generalization task dis- cussed in Sec. 4.2 We present volume renderings (Max 2002) of unseen ensemble members from both the Cloverleaf3D and Nyx datasets. For the unstructured mesh dataset MPAS-Ocean, visu- alize a depth slice of the temperature field over the globe.
C.I CLOVERLEAF3D VISUALIZATION
As shown in Fig. 4 (top), the visualization for the Cloverleaf3D dataset confirms DRR-Net's superior fidelity. Our method accurately captures the complex, high-energy structures of the simulation. In contrast, these fine details are poorly resolved by competing methods, highlighting DRR-Net's effectiveness in representing high-frequency features.
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Ground Truth
DRR-Net (Ours)
PSNR 37.28
SSIM 0.9638
Timel: 0.35g
Params]]: 4.64M
Explorable-INR
PSNR 23.09
SSIM': 0.8917
Timel: 0.24s
Params.:: 14.72M
FA-INR
PSNR
SSIM
Time.: 3.01s
Params 0.97M
Figure 5: Qualitative comparison on the unstructured MPAS-Ocean dataset with one depth layer extracted. The visualization highlights the effectiveness of the DRR paradigm: while the standard embedding- Explorablc-INT suffers from severe artifacts, our DRR-Net successfully captures fine temperature variations, demonstrating the framework's ' power enhance grid-based backbone on challenging, non-Cartesian data, while maintaining efficiency.
C.2 Nyx VISUALIZATION
Fig. (bottom) presents qualitative comparison for the Nyx dataset, focusing high-density gion of interest containing halo features. Both DRR-Nct and the high-fidelity baseline FA-INR pro- vide the most faithful reconstructions of the complex filamentary structures. However, DRR-Net's result can be perceptually closer to the ground truth to minor extent, an observation corroborated by its superior PSNR and SSIM scores.
Beyond visual fidelity, this comparison underscores DRR-Net's significant efficiency advantage. Reconstructing the full 2563 field takes 0.43 seconds, making it over 22x faster than FA-INR. This level of performance is critical for practical scientific applications, enabling interactive data explo- ration. Furthermore, this speed is achieved without any hardware-specifie optimizations. We project that by incorporating low-level acceleration techniques, such as the customized CUDA kernels used modern INRs Muüller al. 2022), the DRR paradigm could support real-time analysis of ensem- ble simulations. This would empower domain scientists with truly interactive workflows, such as iterative volume visualization and notebook-stye -sylle exploratory analysis, for rapid hypothesis testing.
C.3 MPAS-OCEAN VISUALIZATION
To understand the fidelity of reconstructed data for MPAS-Ocean. extract single layer from the depth or radius dimension of the mesh. This layer contains the highest-frequeng features, characterized by the greatest temperature variations across the sphere, from one of the ensemble members. Presented in Fig. 5 we observe that both DRR-Net and the MLP-based FA-INR produce high-fidelig reconstructions that visually close to the ground truth.
The more insightful comparison, however, is between DRR-Net and the standard embedding- based Explorable-INe Both models share similar grid-based backbone, which is inherently challenged
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by the unstructured mesh of the MPAS- -eeann simulation. Despite this shared disadvantage, DRR- Net successfully captures the fine-grained and wavy temperature variations in green, particularly in the zoomed-in view. In contrast, Explorable-INE fails to resolve these details, resulting in a blurry reconstruction. This significant difference in quality demonstrates that the DRR framework is highly effective enhancing the representational power of the underlying embedding structure, enabling to overcome the challenges of non-Cartesian data. Furthermore, while DRR-Net's 's visual fidelity is comparable FA-INR, it achieves this result over 8x faster, showcasing superior balance between reconstruction quality and inference speed.
D CONDITIONAL GENERALIZATION ON LOW-RESOLUTION FIELDS
Table 7: Conditional generalization results for models trained on data with x2 reduced spatial res- olution. Consistent with findings in Sec. 4.3 DRR-Net achieves the highest fidelity and efficiency. Notably, DRR-Net exhibits significant expressiveness advantage in modeling the Nyx dataset in the low-resolution setup.
Unscen Fields
Trained Fields
Inference Time (sec)
Datasct
Model
Params
(Rel / PSNR1 (SIM)))
(Rel / PSNR SSIM))
K- K-Planes
1.49e-01 / 29 44 0.667
1.46e-01 / 99.91 / /0.688
11.4s
3.9M
FA-INR
2.78e-02/ 44.00 0.987
2.76e-02 44.38/ / 00.988
156.3s
9.5M
Nyx
Explorable-INR
4.46e-02 39.89 0.979
3.29e-02 42.86/ 0..84
6.5s
10.0M
DRR-Net (ours)
1.1 / 51.95 0,998
1.15e-02 51.99 / 0.998
8.0s
8.7M
K-Planes
4.33e-01 35.32 0.903
4.15e-01 35.94 0.909
16.15
3.1M
FA-INR
9.92e-02/ 48. 12/12/ 0.991
6.24e-02/ / 52.39 / 00.995
53.8s
1.0
Cloverleaf3D
Explorable-INR
1.17e-01 / /6.65/ 0.989
7,48e-02/ / 50.82 / 00.993
6.2s
4.0M
DRR-Net (ours)
9.38c-02/ 48.60 0,991
5,96e-02 52.80 / 0.995
6.1s
0.9M
While the spatio-conditional generalization task in Sec. 4.3 focused on super-resolution, which in- volved training on x2 downsampled data and evaluating on unseen full-resolution fields, here we evaluate the models directly the low-resolution domain. Since the reduced resolution can sim- plify or even eliminate fine-grained high-frequency features, this theoretically presents a simpler learning objective. Consequently, evaluating the surrogate model in this low-resolution setup serves as a valuable case study to probe the model's capability to accurately capture simpler simulation scenarios.
The quantitative results, presented in Tab. 7 demonstrate that DRR-Net achieves state-of-th of-tthe-rtt fidelity across all metrics. On the Nyx dataset, our method significantly outperforms the nearest competitor, FA-INR, reducing the relative L2 error on unseen fields by approximately 60% (1.11e-02 vs. 2.78e-02) and providing a substantial boost in PSNR from 44.00 dB to 51.95 dB. Notably, DRR- Net's accuracy in modeling the 128 low-resolution Nyx data is significantly higher than in the high-resolution scenario (Ro 42 dB), confirming the substantial representational capacity afforded by the proposed DRR architectural paradigm, the combination of refiner networks and embedding structures. A similar trend is observed on Cloverleaf3D, where DRR-Net consistently attains the highest fidelity for both trained and unseen fields. Crucially, this performance is achieved with high efficiency; DRR-Net maintains inference latency comparable to the lightweight Explorable-INN (s) 6-8 seconds) and is significantly faster than FA-INR, offering inference speedups of approximately 9x on Cloverleaf3D and 19x on Nyx.
E ABLATION STUDY: EFFECT OF EMBEDDING REFINEMENT
To isolate the effect of the non-linear refinement from the DRR formulation, conduct an ablation study, starting with an Embedding-One Only baseline and progressively adding our refiner modules, including the parameter-free functions and the refiner network. The results, presented in Tab. demonstrate the contribution of each component.
Spatial Refiner (SR). The SR provides the most significant and consistent performance gain. Across all three datasets, incorporating the spatial refiner boosts the PSNR of the embedding baseline by large margin, ranging from +2.1 dB Nyx +3.2 dB MPAS-Ocean. This result pro-
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Table 8: Ablation study on the DRR components, demonstrating that the embedding refinement
procedure provides considerable performance gains over the embedding-only baseline.
TFLOPs/
Dataset
DRR-Net Variants
Re4
PSNR1
SSIMT
10 pts
Paramms
Embedding- -nali
4.02e-02
42.64
0,979
40.1
8. 8.1M
Nyx
+ Sputial Retiner
3.17e-02
44.71
57.2
8.7M
Condition Refiner
3.18c-02
44.69
0.986
57.2
8.9M
Embedding--ol
1.24e-02
45.22
25.8
3,57M
MPAS-Ocean
Sputial Retiner
8.54c-03
48.43
47.3
4.57M
Condition Refiner
7.76e-03
49.26
47.3
4.64M
Embedding--Onl
1.43e-01
45.44
0.987
40.1
0.4M
Cloverleaf3D
Spatial Refiner
1.05c-01
48.11
0.992
52.6
0.8M
Condition Refiner
9.81e-02
48.69
0.994
52.6
0.9M
vides clear evidence for the fundamental effectiveness of the DRR paradigm in promoting more expressive spatial representation than can be achieved with embeddings alone.
Condition Refiner (CR). The addition of the CR constitutes the full DRR-Net and further improves performance on the SPPAS-Ocear and Cloverleaf3D datasets by +0.83 dB and +0.58 dB, respec- tively. This demonstrates the value of explicitly refining the conditional features for these more complex ensemble problems. Interestingly, on the Nyx dataset, the CR does not yield an additional gain. We hypothesize this is due to the relatively simple conditional space of this dataset, where the multi-scale base embeddings are already sufficient to capture the inter-parameter variations.
Overall, this study confirms that both spatial and condition embedding refinement components crucial for achieving state-of-the-art performance. The MPAS-Ocean results are particularly telling: the massive performance jump over the Embedding- -nly baseline demonstrates that our refiners are highly effective at compensating for the structural mismatch between a Cartesian base grid and an unstructured dataset, significantly enhancing the model's representational power.
F ABLATION STUDY: NON-PARAMETRIC REFINEMENT WITH π FUNCTIONS
The parameter-free preprocessing function, म, pivotal component of the DRR framework, de- signed to generate higher-capacity input representation without increasing the model's parameter count. In this section, we analyze the efficacy of π r-transformations (P.E. and Structural Super- Resolution) both an isolated non-parametric refinement strategy and pre-conditioner for the refiner network.
We categorize the results into distinct regimes: (1) The Low-Capacity Regime, involving the original low-resolution base embeddings; and (2) The High-Capacie Regime, involving the я- transformed embeddings. This comparative analysis in Sec. F.I sheds light on the critical role of the non-parees panaamaettticc π functions: they not only enhance the embedding structure on their own as a powerful non- par-marrettric realization of DRR but also serve as a necessary utility enable the learning capability of the learnable refiner network. Finally, in (3) Hyperparameter Studies on Full DeR-Net, we vary hyperparameters of π to analyze their impact on fidelity Sec. F.2 and Sec.
F.1 EFFECT OF FUNCTIONS FOR DRR
Tab. quantiffes the impact of the preprocessing function by comparing model performance in its absence versus its presence. This comparison highlights two critical roles of the function: serving performant refinement function and ensuring deep refinement as well as generalization with the refiner.
(1) The Necessity of for Generalization. In the absence of ("No Refinement"), the refiner is forced to operate on the base embeddings in their native state. This presents two critical bottlenecks: first, the base structure implies highly constrained feature space (e.g., the spatial branch yields concatenated embedding vector of only dimension 8); second, the coarse resolution of the original grids provides sparse set of unique feature vectors, resulting in inadequate learning signals for
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Table Ablation study analyzing the impact of the refiner on low-capacity base embeddings versus π- n-trrasformed embeddings. Accuracy metrics are reported for the unseen simulation conditions. We categorize the results into two groups, using the base embedding and the r-transformed embed- ding their respective baselines. The results demonstrate that the proposed r-transformattio not only enhances embedding capacity on its own but also acts as a pivotal enabler, unlocking the full potential of the refiner network.
TFL.PP/ 10 pts 4
Dataset
Refinement Regime
Rel
PSNR1
SSIM
# Params
No Refinement
4.02e-02
42.64
0.979
40.1
8.1M
Refiners
4.48e-02
41.70
0.975
40.1
8.3M
Nyx
T Refinement
4,04c-02
42.60
0.978
55.9
8.1M
Refiners
3.18e-02
44.69
0.986
57.2
8.9M
No Refinement
1.24e-02
45.22
25.8
3.57M
Refiners
1.38c-02
44.29
25.8
3.65M
MPAS-Ocean
Refinement
1.0lc-02
47.00
45.8
3.57M
Refiners
7.76c-03
49.26
47.3
4.64M
No Refinement
1.43-001
45.44
0.987
40.1
0.37M
Refiners
1.62c-01
44.32
0.985
40.1
0.41M
Cloverleaf3D
Refinement
1.13e-01
47.48
0.991
51.7
0.37M
Refiners
9.81e-02
48.69
0.994
52.6
0.82M
optimizing a deep network. Consequently, we observe that adding refiner in this regime degrades performance on unseen members; for instance, the PSNR on the Nyx dataset drops from 42.64 dB to 41.70 dB. Ultimately, the refiner is starved of sufficient data points and representational capacity in this base regime, causing it to overfit to the constrained features rather than learning generalizable data patterns.
(2) π as an Enabler for Refinement. Introducing the π function fundamentally alters the learning dynamic, yielding two distinct benefits.
First, we observe the efficacy of non- non-parametric transformations within the DRR paradigm. Com- paring the "x Refinement" rows to the "No Refinement" baselines in Tab. P reveals that the л-- transformation alone consistently improves fidelity. This demonstrates that the structural priors introduced by n, specifically the spatial smoothness induced by super-resolution and the non-linear multi-scale frequency expansion provided by P.E.. yield immediate representational benefits. This establishes the inherent value of our core contribution, the DRR paradigm: even in the absence of additional learnable parameters, decoupling the inference structure via deterministic refinement produces significantly more expressive representation than the base embedding structure alone.
Second, and more critically, we identify synergistic enablement of the learnable refiner. By pro- jecting the sparse base embeddings into a high-dimensional, high-resolution manifold, æ provides the rich abundance of unique feature vectors necessary to optimize deep neural network. In this ex- panded regime, the refiner no longer bottlenecked by data scarcity. Instead, successfully exploits the increased capacity to deliver significant additive gains over the t-only bascline ("* Refinement + Refiners"). For example, on the MPAS-Oceal dataset, the refiner reduces the relative L2 error from 1.01e-02 (nr-nl to 7. 7.76e-03 (Full DRR-Net). Thus, the results reveal that is not merely preprocessing step, but the foundational enabler that allows the refiner to transition from overfitting to learning generalized, state-of-the-art representations.
For a qualitative comparison illustrating how the learnable refiner overcomes the specific limitations of the r-only regime, such high-frequency noise and aliasing, please refer the visual analysis in Sec. G
F.2 FEATURE UPSAMPLING HYPERPARAMETER: P.E. FREQUENCIES
We study the effect of the level of frequencies used in P.E. proposed by Mildenhall et al. (2020) for feature upsampling, with results shown in Fig. 6 The charts reveal consistent pattern across all datasets: performance steadily increases as the number of frequencies rises to a moderate level, around 6 to 8, after which it either plateaus or degrades sharply.
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Rys R.b. Prequencces Ablation
HPGS Occom: RS. Prequences AMation
ClaveloaftD Kt. Proqueeies Ablation
11.0
PMh
10.7
1115
4000
4530
Prossencio
Figure 6: Ablation studies of the frequency hyperparameter of Positional Encoding for feature up- sampling. moderate frequency level around 6 is observed to consistently improve the efficacy of DRR-Net.
This trend supports hypothesis that the number of P.E. frequencies controls critical trade-off. Initially, increasing the frequency upsamples the features into a higher-dimensions space. This provides the refiner with richer representation, giving it more capacity capture the complex, multi-fre patterns present in the data.
However, this benefit becomes counterproductiiv beyond optimal point for two primary reasons. First, an excessively high-frequency feature space may introduce signal bias that does not match the intrinsic properties of the data with predominantly lower frequency patterns, leading to overfit- ting or aliasing, which is also reported by Zhang et al. (2024). Second, the feature dimensionality can become too high to be optimized effectively given finite amount of training data, creating more difficult learning problem.
Summarizing from the results, a low to moderate frequency level from 2 to for the P.E. is expected to be a safe choice and effectively lift the embeddings to richer multi-frequency representation for the refiner to operate on. On the other hand, an aggressive frequency level can potentially decrease the performance, which is only advisable for datasets known to contain very high-frequency features.
Mys: Resolution Abtation
MPAS MPAS--ccam: Resotation Ablation
Clover 33: Reseluzion Ablation
11
21.0
+000
135
Figure 7: Comparisons of DRR-Net with different maximum resolution for the structural super- resolution (SSR). The baseline configuration without SSR is presented. SSR exhibits the most ad- vantage for MPAS-Ocean and Cloverleai3D.
F.3 STRUCTURAL RESEER-RSOOLUTINN HYPERPARAMETER MAXIMUM RESOLUTIONS
We conduct an ablation study to investigate the impact of the maximum resolution used in Structural Super-Resole Resolution (SSR), with results presented in Fig. Our findings indicate that the effectiveness of the farameter-free SSR can be dataset- oftering significant benefits in some cases while being less effective in others.
For the MPAS-Ocean and Cloverleaf3D datasets, applying SSR yields clear performance gains over the baseline configuration. As shown the charts, an appropriately tuned maximum resolution, such as 152 for MPAS-Ocens and 128 for Cloverleaf3D, leads notable improvement in both PSNR and SSIM over the baseline without SSR. In contrast, for the Nyx dataset, tasking the refiner with learning from an interpolated high-resolution grid can slightly degrade quality compared to learning from the original grids directly.
This divergence appears stem from the interaction between interpolation and the signal's 'ss intrinsic properties. For datasets like MPAS-Ocear and Cloverleaf3D, the underlying signal is relatively coarse and spatially smooth outside of small regions of interest. In these cases, interpolated grid
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DRR-Net g Rv
DRR-Net
π-Only Model
Ground Truth
g
g +
Visuelization Fidelity:
PSNRT: 31.16
SSIM 0.9615
PSNRR:
SSIM!:
PSNRT: 27.38
SSIM": 0.0.0: 0.8822
Model Emficiency:
Timel: 0.05s
Params.. 0.E3M
Timel: 0.04s
Paramsj: 0.85M
Time.; 0.0Ms
Params]: 0.37N
beeand
Visualization Model Effclency:
SSIM":
PSNR
SSIM
Timel:
8.9M
Params];
Time.:
Figure 8: Visual ablation on unseen Cloverleaf3D (top) and Nyx (bottom) members with volume rendering for understanding the role of refiner the proposed DRR paradigm. Columns: (1) Ground Truth; (2) Full DRR-Net: (3) Inference with refiner disabled; (4) m- T-Refinement baseline. Two key observations: (1) Disabling the refiner yields severe artifacts (Col. 3), confirming an integral part of the learned representation in DRR-Net. (2) Compared to the model trained only with the и refinement (Col. 4), the Full DRR-Net (Col. 2) effectively suppresses noisy artifacts smooth regions while sharply resolving high- gighfrreuuncy details in shades of red, validating its superior multi- scale feature fusion.
provides reasonable approximation of the ground truth.
The refiner's task is then manageable
problem of adding high-frequency details to strong baseline.
The Nyx data presents different scenario. Although Fig. F only renders the high-density regions, the full data field is dense with complex. high-frequency patterns even for the regions with lower dark matter density values. coarse grid of such a signal a poor representation that loses significant information. The subsequent upsampling by interpolation acts as low-pass filter and creates smooth field that is an inaccurate approximation difficult, ill posedd learning problem of attempting reconstruct vast amount of lost high- frequency detail. This task can be less effective than learning directly from the original multi-resolume grids that do not suffer from this interpolation-induee information loss.
the true signal. This presents the refiner with a
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G UNDERSTANDING THE EFFECT OF REFINER WITH VISUALIZATION
To complement our quantitative ablation studies on the proposed DRR components, including the complete refinement components (Sec. D and non-parametric refinement functions (Sec. F.I... we present a visual analysis in Fig. 8 to provide a holistic understanding of the refiner network's s role in enhancing representation quality. We visualize unseen members from the Cloverleaf3D (top
row) and Nyx (bottom row) datasets with the volume rendering algorithm [Max (002) across four configurations: (1) Ground Truth; (2) the Full DRR-Net: (3) the Full DRR-Net with the refiner disabled at inference time; and (4) baseline trained with x r-Refinement only. This comparison yields two prominent observations regarding the nature and quality of the learned representation.
Visual Validation of Synergistic Division of Labor. The third column displays the inference result of a fully trained DRR-Net when the refiner module is bypassed. The resulting images consist almost entirely of unintelligible artifacts. This result visually confirms the synergistic division of labor hypothesis proposed in Sec. [32.2] It demonstrates that the base embedding 9 and the refiner Ry become tightly coupled during optimization: the base embeddings have specialized to encode a latent pre-conditioner, likely capturing high-frequency local properties, that relies on the refiner resolve global consistency and feature fusion. Consequently, the refiner is not redundant post- processing step, but an integral component required to decode this specialized base representation into a valid physical field.
Improved Multi-Scale Feature Fusion. A direct comparison between the Full DRR-Net (sec- ond column) and the T-Only baseline (fourth column) highlights the refiner's superior capability in multi-scale feature fusion. The x x-Only model, relying on deterministic upsampling, exhibits two characteristic failure modes: it generates spurious high-frequency noise, manifesting as jagged grid artifacts, in spatially smooth regions, such as the peripheral areas of the Cloverleaf3D simulation. Si- multaneously, it produces over-blurry reconstructions in regions containing genuine high- nigh-freuuecy details, such as the central high-energy structures.
In contrast, the Full DRR-Net successfully acts as a selective fusion mechanism. It suppresses noisy representation in smooth areas while preserving and sharpening true high-frequency signals, as ob- served in the thin high-energy structures from the zoomed-in view of Cloverleaf3D. This validates the refiner's 's capacity to distinguish between aliasing artifacts and physical features, resulting in construction that both smoother in homogeneous regions and sharper at figh-freuuency structures.
H ABLATION STUDY: DATA AUGMENTATION HYPERPARAMETERS
The Variational Pairs (VP) data augmentation strategy, introduced in Sec. 3.4 is general tool for improving the generalization of INR surrogates from the perspective of the data. Its effectiveness can be dependent on the tuning of its noise generation hyperparameters. This section provides detailed ablation study of these parameters to offer practical guidance for their use. Our study focuses on the noise truncation threshold, т, which governs the aggressiveness of the VP augmentation. The standard deviation of the noise, a, is kept proportional to this threshold, making т the primary hyperparameter controlling the trade-off between data diversity and sample reliability. All studies are conducted using our DRR-Net on the Nyx dataset.
We analyze the impact of the threshold on three augmentation variants:
VP-S: The baseline spatial-only augmentation, where the spatial threshold is kept at fixed, conservative value based on the grid resolution.
VP-C: A conditional-only variant designed to isolate the effect of condition noise. This variant was not presented in Sec. and is used here for analytical purposes.
VP-SC: The full spatio-conditional augmentation.
H.1 VP-S: COORDINATE NOISE THRESHOLD
The ablation study for the spatial-only augmentation, VP-S. is presented in Fig. (left). The tested coordinate noise thresholds are derived from the vertex spacing of hypothetical grid resolutions
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VP-5 Threshold Ablation
VP-C Threshold Ablation
45.30
PSM 550
0,995
PSR
0.005
SSH
0.995
45.00
FAR
44.30
-09864
44.00
0.9363
44.00
04:35
46.70
6854
44.30
0.0032
7.83-00
1330633
1,36e-03 03
2.00
2.00-004
6.50a-24
2.50e-33
Coordinate Noise Threshold
Condition Nosse Threshold
Figure 9: Comparison of the noise threshold for the isolated spatial-only and condition-only VP methods in the Nyx dataset. We observe that incorporating VP-S VP-C can both improve the accuracy over a non-augmented baseline.
finer than the native 256 grid with threshold 3.92e-03. A smaller threshold value on the chart, such as 1.30e-03, corresponds to finer hypothetical grid and thus represents a more conservative and constrained noise range. We observe clear trend where performance peaks at this conservative threshold and declines as the noise range becomes either too small or too large.
An optimally chosen threshold provides a performance improvement of approximately 0.2 dB in PSNR over the non-augmented baseline. This result shows that a conservative and constrained noise range can effectively improve DRR-Net for the Nyx dataset. Since VP generates new values via interpolation, limiting the perturbation range is critical. constrained range ensures the augmented samples remain in a region where the simple interpolation function remains reliable approximation of the true data patterns.
Conversely, the performance degradation at higher noise thresholds indicates that allowing aug- mented samples to span broader spatial regions can exacerbate the distribution mismatch between the interpolated values and the ground truth. This introduces unreliable training signals and harms performance. Therefore. it is advisable to keep the spatial noise constrained to relatively small range, such as one-quarter of the native grid spacing.
H.2 VP-C: CONDITION NOISE THRESHOLD
Defining meaningful noise threshold for conditional parameters is more nuanced than for regular spatial grids due to the irregular sampling of the parameter space. To establish principled range of conservative thresholds, first analyze the distances for each simulation parameter across all en- semble members individually. For each parameter, we sort all sample values and find the minimum distance between any adjacent points. This gives set of minimum distances, parameter dimension. We then define our experimental range by taking the minimum and maximum of these values and uniformly sampling five thresholds within this range. This procedure allows us to test spectrum of conservative noise levels, from one based on the tightest sample packing in any dimension to one based on the loosest.
for each
The results for the conditional-on -nly augmentation, VP-C, are presented in Figure 9 (right). The trend is remarkably similar to that of VP-S, confirming that conditional augmentation can effectively boost model performance. Performance peaks at a moderate threshold of 1. 1.e003 and is worse on either side of the magnitude range, with the lowest and highest thresholds performing below the non-augmented baseline.
These results imply the same principle observed in VP-S: a sufficiently constrained noise level can provide diverse learning signals beyond the original sparse parameter samples, while still ensuring the reliability of the conditional interpolation. However, since sampling strategies for conditional parameters can vary significantly between simulations and may not resemble regular grid, de- vising a reasonable noise level can be more challenging and may benefit from a domain- -spciffc understanding of the parameter space.
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VP-SC Threshold Ablation
44
0.9856
44
98955
PSNR
PSNR SSIM
-0.8554
SSIM
0.9853
0.9652
44
5.0... PANRI AAIAR
W.W.M....... SRIM
2.00c-04
2.00e-03
Condition Noise Threshold (Constant Coord Noise)
Ablation study on varying the condition noise thresholds for VP-SC with a constant spatial noise level. Results show that adding the condition noises consistently improve DRR- Net quality marginally, compared to VP-S.
Figure 10:
H.3 VP-SC:
NOISE THRESHOLD
Finally, we analyze the full spatio-conditional augmentation, VP-SC. In this study, presented in [Og[TO] we vary the conditional noise threshold using the same range the VP-C experiment, while keeping the spatial noise constant threshold 3.92e-03. The spatial noise level also the setup for the main evaluations in Sec 4.2 2 and Sec.
The results show that adding conditional noise on top of spatial noise provides consistent, though marginal, performance improvement over using only spatial augmentation. A key observation is that the model is much more tolerant to the choice of the conditional threshold in this combined setting. Unlike the isolated VP-C study, all tested threshold values in the VP-SC configuration outperform the VP-S baseline. This suggests that the presence of spatial noise may have a regularizing effect that makes the model more robust to variations in the conditional augmentation.
H.4 FINAL REMARKS ON VP AUGMENTATION
Our ablation studies the Variational Pairs augmentation strategy reveal several key takeaways for practical application.
Spatial-only augmentation (VP-S) serves as strong and reliable baseline. For simulations on reg- ular grids, its noise threshold is straightforward to configure, requiring minimal hyperparameter tuning. Conditional-only augmentation (VP-C) can also be effective, but determining an optimal threshold is more challenging due to the irregular nature of parameter spaces and may benefit from domain-specific knowledge.
The full spatio-conditional variant (VP-SC) offers path to the best augmentation result. It can provide an additive improvement over VP-S, a pattern observed across models Sec. [..4. Fur- thermore, our study here shows that VP-SC is more tolerant to the specific choice of the conditional noise threshold. Therefore, we recommend VP-S as a simple and robust starting point, with VP- SC more advanced option to extract further performance, especially in scenarios where tuning conditional noise complex.
I APPLICATION TO VISION AND GRAPHICS TASKS
To demonstrate the versatility of the DRR paradigm, extend its application beyond scientific simulations to common vision and graphics tasks. This section evaluates DRR-enhanced INRs for neural image representation (Sec. [.I) and Neural Radiance Fields (Sec. 1.2)
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Table 10: Performance comparison of various INRs with and without DRR for two Gigapixel im- ages. Rows marked "+ DRR" show the change (A) relative to the base model above. Green indicates an improvement. Strong baseline models, including NGP and SIREN, are also evaluated to serve as reference.
Inference Time (sec)
Gigapixel Image
Model
PSNR1
SSIM
LPIPS_
# Params
Grid
25.80
0.9180
0.2504
0.86s
4.2M
+ DRR
+0.12
+0.0016
-0.0040
1.16s
4.3M
Girl with a Pearl Earring
MRGrid
25.86
0,9191
0.2468
1.17s
4.2M
+ DRR
-0.02
-0,0012
+0.0055
2.28s
4.3M
23466x20000
NGP
26.45
0,9154
0.2235
0.90s
5.2M
SIREN
24.64
0.8713
0.2827
27.82s
2.1M
Grid
27.13
0.9396
0.1700
2.13s
4.2M
+ DRR
+1.11
+0.0108
-0.0164
2.58s
4.3M
Tokyo
MRGrid
28.08
0.9493
0.1476
2.94s
4.2M
+ DRR
+0.92
+0.0070
-0.0119
5.55s
4.3M
21450x56718
NGP SIREN
25.63
0.8978
0.1873
2.22s
5.2M
14.31
0.1061
0.7288
71.638
2.1M
1.1 GIGAPIXEL IMAGE REPRESENTATION
Experimental Setup We evaluate the DRR paradigm on the task of gigapixel image representa- tion, where INR is trained map 2D pixel coordinates their corresponding RGB values. We apply DRR to two embedding-basee backbones: singlc-resolution feature grid (Grid) and a multi- resolution grid (MRGrid). The models contain a feature grid encoder and 2-layer lightweight MLP decoder for prediction. Both structural super-resolution and P.E. feature upsampling are ap- plied prior to the refinement. These DRR-enhanced models are compared against their original counterparts. Although the experiment aims to study the effect of the DRR paradigm on different embedding- models, we present two effective baselines, NGP (Müller et al. (0022) and SIREN (Sizzmann et al., 2020)) to provide performance contexts. The evaluation is conducted ages with different characteristics: the relatively smooth "Girl with Pearl Earring" at 23466 x 20000 resolution and the more challenging "Tokyo" cityscape with significantly more objects and textures at the resolution of 21450 x 56718.
two im-
Quantitative Evaluation. The results in Tab. 10 show that DRR substantially improve construction quality, particularly for images with complex, high-frequency details. On the visually intricate "Tokyo" cityscape. applying DRR provides significant performance boost both base models. improves the PSNR by +1.11 dB for the single-resolution Grid and +0.92 dB for the Musti-Resolution Grid (MRGrid). We later show that this creates perceptually significant improve- ment of the reconstruction in the visualizations. Notably, both DRR- DRR-enhanced models considerably outperform the strong NGP baseline this challenging image.
The results on the "Girl with Pearl Earring" image, which contains smoother content, are more nuanced. While DRR provides modest improvement to the simple Grid model, it slightly degrades the performance of the more powerful MRGrid. This suggests the benefit of the DRR refiner is most pronounced when the base representation has room for improvement or when the signal is highly complex. For simpler signals where a strong base model like MRGrid may already be sufficient to capture the content, the additional capacity of the refiner might not provide significant benefit.
Qualitative Evaluation. Fig[] showing cropped views at progressive levels detail. The visualization studies the effect DRR on the MRGrid model and includes the NGP model for reference. The results reveal significant visual improvement offered by the DRR paradigm.
presents qualitative comparison the complex "Tokyo" image,
The first evidence of reconstruction loss is the level of image noise. NGP exhibits the most noise, creating jagged and inconsistent textures, followed by the base MRGrid. The improvement from DRR is evident at moderate magnification from the level 2 images in the second column,
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Detail Level
Detail - Level
Detail Leve!
Ground Truth
PSNR1: SSIVI:
PSNR
SSIHI:
PSNRT SSIM 0.264
MRGrid + DRR
PSNRT: 25.00 S9.7
LPIPSL: 0.260
PSNRT: 25.17 SEI 0.605 LPIPS 0.294
PSNRI: 26.15 S50M1: LPIPS]: 081
MRGrid
LP
PSNRT:
NGP
Visualization Gigapixel image with a highly complex object composition pro- gressive level of detail. The addition of DRR significantly improves the visual fidelity for the Multi- Resolution Grid (MRGrid) model. "Tokyo" by Trevor Dobson (CC BY-NC-ND 2.0).
Figure 11:
where the DRR-enhanced image smoother without losing structural details. Most importantly, DRR demonstrates a superior ability to faithfully recover critical object details. In the highest mag- nification view level 3, the traffic signs poorly reconstructed by both NGP and MRGrid. In the NGP prediction, the signs nearly disappear into blurry artifacts, while the MRGrid struggles to re- cover defining features like the red diagonal line. The DRR-enhanced MRGrid, however, preserves the visual characteristics of the signs to much greater extent, allowing them to be casily perceived and recognized.
This highlights the advantage of DRR, especially for challenging data with densely packed, high- frequency features across the spatial domain. The visual results confirm that DRR provides a sig- nificant additive benefit to embedding based structures, improving reconstruction fidelity while pre- serving their critical efficiency properties. The improvement is most pronounced when a gaid model is applied to suitable dataset, such complex scene with diverse features throughout the domain for dense feature grid.
This success suggests promising direction for future work. For data with sparser features where hash-grid models like NGP excel, it would be an interesting area exploration to study whether further.
applying the DRR principle to refine the hash grid could boost performance
Zero-Shot Super-Resolution Evaluation. To further test the quality of DRR-enhanced models, evaluate the models the more challenging task of zero-shot x2 super-resolution. Similar to our spatio-conditional evaluation, the INRs are trained on data that have been downsampled to half their original resolution. The models then evaluated their ability reconstruct the original,
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Table 11: Zero-shot x2 super-resolution evaluation for Gigapixel images. The results for DRR- enhanced models are shown the A compared to the baseline in the previous row. NGP and SIREN are shown as performance references. Green indicates an improvement.
Inference Time (sec)
Gigapixel Image
Model
PSNRT
SSIM
LPIPS!
# Params
Grid
25.80
0.9178
0.2514
0.88s
4.2M
+ DRR
+0.13
+0.0021
-0.0053
1.16s
4.3M
Girl with Pearl Earring
MRGrid
25.86
0.9188
0.2489
1.19s
4.2M
DRR
+0.03
+0.0003
+0.0009
3.21s
4.3M
Train: 11733 10000
NGP
26.31
0.9151
0.2354
0.90s
5.2M
Test: 23466x20000
SIREN
24.81
0.8776
0.2786
27.82s
2.1M
Grid
27.15
0.9398
0.1727
2.12s
4.2M
+ DRR
+1.11
+0,0106
-0,0162
2.56s
4.3M
Tokyo
MRGrid
28.07
0,9492
0.1519
2.92s
4.2M
+ DRR
+0.73
+0.0057
-0.0096
5.54s
4.3M
Train: 10725 x 27308
Test: 26750
NGP
25.60
0.8975
0.1923
2.238
5.2M
SIREN
26.47
0.9268
0.1903
71.28s
2.1M
full-resolution images, task which requires the learned representation to be truly continuous and generalizable across scales.
The results, presented in Tab. demonstrate that applying DRR consistently enhances the zero- shot super-resolution capabilities of both base models. The overall observations consistent with the previous results with full-resolution training in Tab. [0] where DRR substantially boosts the per- formance of models for the complex "Tokyo" scene. For the "Girl with a Pearl Earring" image, DRR also provides a consistent, though more modest, improvement for both backbones. Notably, for the MRGrid model, DRR provides slight improvement in this more challenging task, in contrast the slight degradation seen in the standard reconstruction setting. These results consolidate the finding that DRR provides a consistent fidelity advantage embedding-baat models while maintaining promising inference efficiency.
As a final remark of this evaluation, the similarity of these outcomes the standard evaluation setting shown in Tab. [o]aa suggest that x2 downsampling factor may be too conservative to fully probe the generalization limits of INRs on such ultra-high-resolution images. Future explorations using more aggressive downsampling factors, such as x or x8, could offer deeper insights into the spatial generalization capabilities DRR-enhandn enhanced models.
I.2 NEURAL RADIANCE FIELD FOR NOVEL VIEW SYNTHESIS
Novel View Synthesis (NVS) is another application of INR that enables faithful and view-consistent 3D reconstruction with only 2D supervision. Neural Radiance Field (NeRF) (Mildenhall et al.) 2020 is one seminal INR formulation for NVS, and we demonstrate the effect DRR on NeRF models for two NeRF synthetic datasets.
We integrate DRR with three embedding- based NeRF models. The same models in the Gigapixel image representation tasks are adapted, including the basic 3D feature grid INR (Grid) and a multi- resolution feature grid model (MRGrid). In addition, a plane-based decomposition model (TriPlane (Chan et al. 2022 2) is included as competitive embedding structure for 3D data. For this exper- iment, we apply a simplified version of DRR where the refiner directly enhances the base embed- dings without the preprocessing steps of super-resolution or P.E. feature upsampling. All models are trained and evaluated following the implementation of the open-sourced NerfAce repository (Li al. 2023) We again include two baselines in Tab. T2 references of performance: the original MLP. based NeRF Millenhall et al. 2020) and the Instant-NGP (Müller et al. 2022).
The results in Tab. 12 show consistent trend: applying DRR to models built embedding struc- tures consistently improves both reconstruction (PSNR) and perceptual (LPIPS (Zhang et al. 2018)) metrics. Notably, this performance gain is also accompanied by reduction in inference time for
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Table 12: Performance comparison of neural radiance field (NcRF) models with DRR-Enhanced embedding structures. The performance of NGP (Müller et al. 2022) and the original NeRF model (Mildenhall al. 2020) presented as a reference. Values for DRR" rows show the change rel- ative to the baseline structure above. Green indicates improvement (higher PSNR, lower LPIPS).
Inference Time (scc)
Dataset
Model
PSNR
LPIPS
# Params
Grid
33.33
0.041
48s
33.6M
+ DRR
+0.06
+0.002
51s
33.6M
MRGrid 32.49
0.053
19.0M
Lego
+ DRR
+0.31
-0.001
51s
19.0M
TriPlane
35.32
0.032
71s
9.4M
+ DRR
40.07
-0.005
9.7M
NGP 35,46
0.026
54s
12.6M
NeRF
33.69
0.051
318s
0.6M
Grid
25.31
0.082
48s
33.6M
+ DRR
-0.54
+0.004
53s
33.6M
MRGrid 25.37
0.085
52s
19.0M
Drums
+ DRR
+0.06
-0.001
50s
19.0M
TriPlane
25.06
0.104
9.4M
+ DRR
40.30
-0.018
9.7M
NGP 25.69
0.083
53s
12.6M
NeRF
25.22
0.089
277s
0.6M
MRGrid, as the multiple grids are refined into a single representation with multi-scal features, thereby reducing the computation and overheads involved in the interpolation functions.
Interestingly, we observe that applying DRR can degrade the performance of a simple, single- resolution Grid the Drums scene. Similar the MRGrid observation from Sec. [.]] we hy- pothesize this occurs when a base representation is already performing at its capacity limit for a given task, such as high-resolution feature grid with 33.55 million parameters for single radiance field. In such cases, adding more non-linear transformations via the refiner may not meaningfully extend the already-saturated features. What remains clear, however, is the main trend: for modern, scalable embedding structures like MRGrid and TriPlane, DRR is an effective tool for boosting the performance of the resulting NeRF models with minimal speed penalty.
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Reviews:
[{"review_id": "RgfgPHedJD", "reviewer": "Reviewer_JRMt", "summary": "Implicit Neural Representations (INRs) either give you high accuracy but slow queries, or fast queries with weaker expressivity. This work proposes a Decoupled Representation Refinement (DRR) method, which runs a deep refiner network once offline to bake rich structure into a compact embedding that can be queried efficiently. Authors claim the proposed model, DRR-Net, plus a data augmentation strategy (Variational Pairs), achieves state-of-the-art fidelity on ensemble simulation surrogates while being up to 27x faster than strong high-accuracy baselines and still competitive with the fastest models.", "questions": "Please refer to the weaknesses (Each weakness raises one or more questions).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1) DRR-Net hits a strong pareto point as it produces competitive results compared to baselines like FA-INR in PSNR/SSIM on large 3D scientific ensembles, but delivers ~10x-30x faster inference, and it outperforms fast baselines like Explorable-INR in fidelity while staying in their runtime class.\n\n2) Instead of treating each simulation parameter independently or relying on low-rank factorization, DRR-Net builds unified multi-parameter conditional embeddings and refines them jointly, so nonlinear interactions between parameters are captured and then cached for fast reuse. This is something that is not explored as much in other methods especially the models that are acting as simulators. \n\n3) Experiments are comprehensive and the baselines are relevant and recent (Some missing ablations, see weaknesses).", "weaknesses": "1) The stated one time cheap evaluation of refiner R seems to be dependent on the problem setup, boundaries etc. does that mean if the boundaries have changed a new refiner is required? If yes this seems like a major drawback compared to other methods especially given that training times for this method are relatively longer than most other frameworks. Can the authors clarify whether it is correct, that any specific change in the domain (boundaries etc.) requires retraining? Furthermore is there any assumption on the conditioning parameters? Can these be a field like conditioning and not simply scalars? Do these affect the training? \n\n2) Experiments hold out ensemble members but still operate within the same parameter ranges as training (Table 1). Can you characterize DRR-Net’s behavior under extrapolation? For example, how does it perform for parameter settings outside the training envelope, and how does this compare to FA-INR and Explorable-INR? Currently extrapolation claim is not well justified. \n\n3) Although model parameter counts are provided in the table, it is difficult to establish if the improvements are not originated from variations in capacity and what's the role the parameter counts plays in this. The interplay between the model size and the interpolation/extrapolation capacity is not demonstrated.\n\n4) There is very little on detail on the training procdedure, actual objective/objectives, and hyperparameter setting. This can make the procedure vague for the reader. Similarly although explained in text, this can be hard to follow how and in which order the components are trained for this framework .", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jeRn8BURhD", "reviewer": "Reviewer_eTAQ", "summary": "The paper proposes a decoupled representation refinement for implicit neural representations of large-scale scientific simulations. \nThe decoupled representation refinement first trains a embedding-based model. The embedding, represented by a base structure, can be preprocessed and a refiner network learns an improved representation, utilizing the expressive power of deep neural networks. The updated base structure can be used for inference via a fast query function. \nThe updated base structure can also be cached. Additionally, the authors propose two augmentation strategies. The method is evaluated on three scientific datasets and compared against relevant benchmarks.", "questions": "See questions in weaknesses above.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The method is well motivated and an effective compromise between fast embedding-based models and MLP-based approaches. \n- The paper is well written \n- Detailed experiments on three scientific datasets show quality vs. training/inference speed tradeoff. DRR-Net compares favorably.\n- Easy to implement data augmentation method that improves PSNR/SSIM in most cases\n- Additional experiments on vision/graphics tasks in the appendix", "weaknesses": "- The paper is generally well written, but the description of the condition encoder was a bit unclear to me. How does \"a set of 1D multi-resolution feature lines for each simulation parameter\" support scalability to \"arbitrary parameter dimensions\" Could you please clarify this?\n- Previous approaches have considered higher resolution scientific datasets, e.g., [1]. Can DRR-Net scale to similar resolutions; what changes are necessary?\n- The proposed data augmentations seem effective for interpolation settings of conditional inputs. Have you done any tests for extrapolation of conditional inputs?\n\nOverall, this is a convincing paper to me with strong experiments. Since I haven't been following the recent literature for implicit neural representations closely, it's difficult to fully assess the novelties of the method for me. Therefore I will give a low confidence to my rating. \n\n[1] https://arxiv.org/abs/2308.02494", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KIzURvfFJc", "reviewer": "Reviewer_MCZm", "summary": "The paper suggests a new approach for training and inference of Implicit Neural Representations (INRs), also called Neural Fields (NFs): Decoupled Representation Refinement. \n\nNFs are MLPs with (continuous) coordinates (and potentially additional time or conditioning variables) as inputs. While Nfs are powerful representations, they typically suffer from a tradeoff between reconstruction fidelity, which requires larger, parameter-rich MLPs, and inference speed, which decreases with MLP size. \nTo bypass this tradeoff, DRR aims to decouple the learned high-capacity representation from inference queries, thereby allowing for inference speedups of up to x27. \nThe high-capacity representations are created by a learned refiner, which takes the multi-resolution input embeddings upsampled to the same scale and outputs a dense grid. The Decoder, a small, lightweight MLP, can then be queried quickly by taking the coordinate and the conditioning embedding as features.\n\nAs data augmentation for improved generalization, the authors use a modified version of Variational Coordinates (VCs). Instead of a piecewise linear prior, their augmentations (VP Spatial and VP Spatio-Conditional) impose a smoothness prior, which fits the application domain (scientific simulations) much better.\n\nIn experiments, the authors demonstrate the performance of DRR on 3 datasets with 3-6 dimensional fields: Nyx (a dark matter simulation), MPAS-Ocean (an earth system simulation), and Cloverleaf3D (a hydrodynamics simulation). They show that DRR overcomes the conventional fidelity/efficiency tradeoff by achieving both the good reconstruction quality of large, expensive INRs and the speed of fast, embedding-based methods.\n\nIn the appendix, the authors provide further ablation studies which justify their architecture choices and show the DRR framework also applies to the classic NF domains, such as 2D image superresolution and 5D neural radiance fields.", "questions": "## Major Remarks\n- The paper lacks clarity about the method. Especially in 3.3, I am still not certain what exactly the inputs/outputs and their sizes are for the different components. Figure 2 only helps so much. E.g., what is a 1D multi-resolution feature line? What exactly is the output form of the embeddings? Is it just a concatenated vector of embeddings? Of what dimension?\n- Building on the previous point, what are the memory costs at inference for DRR, and how do they scale? How memory-intensive are the spatial embeddings? How big is the overhead for the embeddings, i.e., how many inference queries are needed to amortize the initial encoder forward pass?\n- Please add some more metrics which indicate simulation quality. For example, in fluid simulations, crucial indicators are the preservation of conservation laws (e.g., momentum or energy), or the accuracy of spectral properties (e.g., kinetic energy spectra) which indicate how well the method captures turbulent or small-scale features.\n- In the field of neural graphics/rendering, serious efforts have been made to enable fast inference of neural graphics, e.g., [1]. I believe that this should at least be in the related work and discussed, e.g., why the same idea works/not work for simulations, and why such strategies are not pursued.\n- You claim, e.g., SIREN (and NGP) is SOTA on Page 21; this is not true. Several SIREN improvements have been published; see [2] for a current comparison. \n\n## Minor Remarks\n- Page 3 Line 129-130: You use \\mathcal{P} for the parameter count and the number of points in a simulated field as P. This reduces readability. Please use something other than P.\n- Page 8 Table 3: You show the performance on trained and unseen fields in the upsampled case. Can you also provide performance metrics of trained vs untrained on the same sampling scheme, i.e., how much performance drop is there in fitting a single field vs the generalization ability of a larger pretrained one?\n- I also feel like the fact that your method also works for NerFs or image superresolution should be highlighted a bit more in the main text; the fact that the method is so general is noteworthy. (Yes, I know the page limit makes things tough).\n\n## Conclusion\nI think the core idea of the method is sound, and while the experimental evaluation could be broader, it is good enough for acceptance, especially given the additional ablations and experiments in the appendix. The reason I currently lean towards rejection is the, in my view, bad/unclear presentation at times, insufficient metrics for numerical simulations being reported, and using baselines that are no longer SoTA as \"SoTA\". If the authors can address my points, I am willing to discuss and adjust my score.\n\n[1] Duckworth, Daniel, et al. \"SMERF: Streamable memory efficient radiance fields for real-time large-scene exploration.\" ACM Transactions on Graphics (TOG) 43.4 (2024): 1-13.\n\n[2] Essakine, Amer, et al. \"Where Do We Stand with Implicit Neural Representations?\" A Technical and Performance Survey 11 (2024).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "## Soundness\nThe experiments support the central idea of the method, i.e., good reconstruction quality and fast inference.\n\n## Contribution\nAchieving fast inference times with NFs is a known problem, and the authors offer an interesting solution that combines a learning-based approach with the advantages of multi-resolution representations, which have shown good results in neural graphics processing.\n\n## General\n- Very interesting take on a known problem of NFs: Large MLPs are slow in inference. The large MLPs use their large parameter count to construct rich internal representations in their hidden layers. Embedding-based methods like NGP encode features in learnable feature grids, which makes them fast but also memory-intensive. DRR proposes a spatial refiner to achieve similar rich encodings to be utilized by a fast decoder.\n- DRR is shown to work across simulations with fields of different output dimensions and also more classic NF applications such as image superresolution and NerFs.\n- Good ablation studies to justify the different architecture choices.", "weaknesses": "My main issue with the paper is that it lacks clarity about the method.  Furthermore, the reported metrics are insufficient for physical simulations. Certain claims regarding “SOTA” also do not stand up to scrutiny.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nVSXPJEgPe", "reviewer": "Reviewer_V9m2", "summary": "The paper explores the usage of INR methods for surrogate modeling. Observing that embedding-based methods lack representational capacity while MLP methods are too slow, they propose an approach in which embeddings are first transformed through a parameter-free interpolation and stacking function before being projected by a complex non-linear mapping. The resulting function is then treated as the embeddings for interpolation. The authors further propose augmentation strategies involving perturbing the training data under continuity assumptions in the data. This is evaluated across three 3D challenges.", "questions": "1. How are the datasets constructed? \n2. How does the proposed approach compare to established methods when hyperparameter tuning is performed on the other methods?\n3. How does the proposed approach compare to standard ML approaches for surrogate modeling like autoregressive models or diffusion models?\n4. What are the full details of the models used in this paper?\n5. What types of problems is this method well suited for? What problems is it poorly suited for?", "limitations": "", "rating": null, "confidence": 3, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "The conflict between efficiency and accuracy is an important problem in surrogate modeling. The paper does a good job of outlining how this is a problem for INR methods and in describing the weaknesses in prior iterations of INR methods that this approach is trying to correct. The datasets used seem interesting and the focus on 3D problems is well-suited to the scalability goals of the paper.", "weaknesses": "Overall, this submission seems to largely apply methods for graphics to scientific tasks with little adjustment for the domains targeted. It seems like an interesting graphics approach, but the submission requires a lot of work before it's a viable paper on surrogate modeling. The text of the paper is missing a lot of information necessary for evaluation and reproducibility. Comparisons largely focus on similar classes of method and not on methods used in the space. Given the lack of information, it's not clear whether the proposed method is actually effective for the given tasks. I can't speak to the novelty of the technique as my expertise is in surrogate modeling rather than image reconstruction. Here are some specific issues and suggestions for improvement:\n\n1. Generally the experiments do not provide sufficient information to evaluate the soundness of the experiments. These manifests in a few ways:\n    1. The datasets here reference codebases rather than preconstructed datasets. Given these are transient problems, what procedure is used to generate the dataset? Are these constant initial conditions are a variety of system coefficients? What time step are these gathered from? Does the system converge to a steady state or is this instead solving a steady state version of the problems? How far away are the parameters? These details are extremely important for evaluating surrogate modeling tasks. \n   2. What grids are actually being used for the grid-based methods in terms of both resolution and dimensionality? Are all grid based methods getting access to the same resolution and dimensionality of embeddings?\n   3. Given the uncertainty over the dataset, its actually not clear that interpolative methods like INR are appropriate for modeling the systems. For instance, ocean simulations over long time horizons can often produce very different results at the same time range based on minor perturbations, though to my knowledge this is not the case for the astro-oriented hydrodynamic simulations. \n2. The baselines do not appear to be well executed. The appendix states that all embedding models were trained with a single learning rate or with exact settings recommended by the original paper. This is not a reasonable approach and will generally promote the \"new\" model by default since that is the model where hyperparameters are tuned for the given datasets. Different models will often perform differently at different hyperparameter ranges on different datasets. Good practice is to do a hyperparameter search for each model independently. \n3. It's not clear that the refiner network is superior to just using denser grids of higher dimension. The refiner network seems to operate entirely on the embedding vectors with no additional information. Apart from potentially enforcing some low-rank structure in $\\hat G$ (which the refiner can learn to ignore), passing them through a complex nonlinear mapping ultimately means you're operating on a grid of embeddings with a certain resolution and dimension with smoothness determined by the interpolation used on the highest resolution grid. While it could turn out that the initial low rank structure is heavily used by the refiner and provides a significant advantage. \n4. Several of the augmentation innovations seem overfit to the data. The data here is largely smooth with very little variation which lends itself well to low order continuity assumptions. However, this is not generally the case for simulation data which often has sharp discontinuities in the form of shocks or interfaces. In parameter space, one of the complexities of physical simulation that researchers are interested in when they're running ensembles of simulations is the existence of bifurcations in parameter space which would likely be hurt by augmentation strategies relying on strong smoothness assumptions. Smoothness is often a reasonable choice, but it's important to also demonstrate cases where the choice may not be appropriate to evaluate whether the proposal is a reasonable default. This concern could  also be addressed by exploring the relationship between the source data and optimal interpolation\n5. Surrogate modeling is an extremely active area of research, but the only baselines are two very recent INR models from a single group and one INR method that seems to have been developed for graphics purposes. Diffusion-based baselines are likely the best fit here since your models map from parameters to a field, though given they're all naturally transient, autoregressive baselines rolling out to the given time frame could also apply. \n6. Metrics - the metrics used are all image reconstruction metrics. Typically, relative error or various forms of normalized RMSE are used in surrogate modeling. Often the appropriate metric depends on the time horizon of the predictions. For the ocean simulations, generally longer time horizons would use smoothed metrics, for instance. Pointwise metrics are often more appropriate for short time horizons. From the appendix, these scenarios seem to be largely empty space which would mean localized metrics around regions of interest are probably more informative than metrics that will be heavily weighted by regions with little activity. \n\nMinor\n1. Figures are not super informative. Figure 1 for instance is mostly a less informative version of figure 2. \n2. Table 3 should list the metrics. It's inferrable from the main text, but tables should be self-contained. \n3. Limitations right now are mostly limited to trivial issues with the implementation rather than more insightful analysis of where this line of research needs to move to address more realistic challenges.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZaNug3QCS1", "reviewer": "Reviewer_AJJD", "summary": "This paper proposes the Decoupled Representation Refinement (DRR) paradigm to track the fundamental trade-off in Implicit Neural Representations (INRs): High-fidelity MLP-based models are computationally expensive and slow to query, while fast embedding-based models lack expressive power. The core is a deep refiner network, which is used to encode rich representations into a compact and efficient embedding structure. Experiments show that the proposed method achieves SOTA fidelity, while being up to 27x faster at inference than high-fidelity baselines and remaining competitive with the fastest models.", "questions": "Please refer to the weaknesses. Overall, DRR is a simple and effective method for improving INR performance. However, the paper lacks clarity on several key issues (Weaknesses 2 and 4). The reviewer looks forward to the authors’ response to these concerns.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method is simple and effective. DRR shows potential for broad application.\n2. Extensive experiments and ablation studies thoroughly validate the effectiveness of the DRR framework and most of key components.", "weaknesses": "1. Variational Pairs (VP) lacks sufficient background in the introduction.\n2. In Synergistic Representation Learning (Line 192), the authors compare DRR with grid-based methods regarding multi-scale representation and feature fusion. Authors claim: “The refiner learns a potent, non-linear function to perform this fusion offline, baking the multi-scale features directly into the cached grid $\\mathcal{G}^{\\prime}$.” The reviewer suggests adding a visual comparison between $\\mathcal{G}^{\\prime}=\\hat{\\mathcal{G}}$ and $\\mathcal{G}^{\\prime}=\\hat{\\mathcal{G}}+\\Delta{\\mathcal{G}}$ to illustrate whether DRR indeed performs multi-scale feature fusion in a more reasonable manner.\n3. As DRR resembles a residual-based method, the reviewer believes discussing analogies to such methods in the introduction and related work would better elucidate significance of DRR for INRs. For instance, the approach described in Equation 3 may enable identity mapping similar to skip connections in ResNet, providing a more flexible forward pathway for complex representations.\n4. Although the authors present ablation studies on the parameter-free preprocessing step, $\\pi$, across different resolutions in Sec. F.1, the reviewer is more interested in experiments without $\\pi$. Specifically, the reviewer would like to know whether $\\mathcal{G}^{\\prime} = \\mathcal{G}+R_{\\psi}(\\mathcal{G})$ remains more effective than $\\mathcal{G}^{\\prime} = \\mathcal{G}$. This experiment would more conclusively demonstrate whether improvements of DRR over other SOTA methods stem from mapping $\\mathcal{G}$ to a higher-dimensional space or from the residual refining module (the core contribution).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)