Pith. sign in

REVIEW 3 major objections 5 minor 59 references

A variational latent neural-field model gives reduced-order PDE forecasts with parameter-space uncertainty and exact conservation by construction.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 08:00 UTC pith:NZRDNPYP

load-bearing objection Solid unification of amortized latent UQ with exact conservation-by-construction; useful for multi-query conservation-law ROMs, with the main caveat already flagged (diagonal amortized GP over parameters only). the 3 major comments →

arxiv 2607.10965 v1 pith:NZRDNPYP submitted 2026-07-13 physics.comp-ph math-phmath.MPphysics.data-anphysics.flu-dyn

Structure-preserving variational neural fields: Uncertainty-quantified reduced-order modeling of nonlinear conservation laws

Aviral Prakash , Marc L. Klasky This is my paper
classification physics.comp-ph math-phmath.MPphysics.data-anphysics.flu-dyn
keywords exact conservationlatent dynamics modelsuncertainty quantificationshock-propagation problemsgenerative modelsscientific machine learningvariational neural fieldsreduced-order modeling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Reduced-order latent dynamics models speed up parameterized simulations of conservation laws, but they usually give only point forecasts and do not guarantee that mass or other conserved quantities stay conserved. This paper builds a single variational latent neural-field framework that does both: it models the latent state with Gaussian-process-inspired surrogates so that predictive confidence rises when a new parameter is far from the training set, and it can embed the solution so every sample (and the mean) lies on the conservation-law manifold. Three variants are compared—purely data-driven, softly physics-penalized, and exactly structure-preserving—on advection, Euler, and shallow-water problems. The structure-preserving version stays accurate under sparse, noisy data while still delivering useful uncertainty bounds for both interpolation and extrapolation. A reader who needs fast multi-query surrogates that stay physically admissible and flag their own ignorance cares because those two properties have almost always been pursued separately.

Core claim

The paper shows that a variational latent neural field with amortized Gaussian-process mean and variance, combined with a decoder that outputs a high-dimensional vector potential of a space-time divergence-free solution-flux field, simultaneously supplies parameter-aware uncertainty and exact nonlinear conservation for reduced-order forecasts of parameterized conservation laws, remaining competitive under sparse and noisy training data.

What carries the argument

ECLEIRS-UQ: the random solution-flux field is realized as the row-wise divergence of a skew-symmetric matrix potential learned by a variational latent neural field, so every sample (and, by commuting expectation and divergence, the mean) is exactly divergence-free.

Load-bearing premise

The method treats the latent posterior as a diagonal Gaussian whose correlations live only across parameters and whose mean and variance are amortized by neural networks; if those surrogates miss strong latent coupling or true out-of-distribution distance, the uncertainty claims fail even when conservation still holds.

What would settle it

On a held-out far-extrapolation parameter for any of the three PDEs, compute whether the reported two-standard-deviation coverage still brackets the true solution while the structure-preserving residual stays at machine precision; systematic under-coverage with rising residual would falsify the joint UQ-plus-conservation claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Multi-query campaigns for parameterized conservation laws can use a single trained model that both accelerates prediction and flags out-of-distribution parameters via elevated latent variance.
  • When training data are sparse or noisy, the exactly conserving decoder remains more accurate than unconstrained or softly penalized alternatives without sacrificing coverage or error-correlation metrics.
  • The mean of sampled solution-flux fields inherits exact conservation, so ensemble statistics remain admissible even under distribution shift.
  • The same amortized GP surrogates can be dropped into static latent operators when pure time-forecasting is not required.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Uncertainty maps concentrated on shocks and wave fronts could serve as acquisition functions for adaptive high-fidelity sampling without extra model changes.
  • Replacing the diagonal GP latent with a richer generative prior would test how much of the reported OOD behavior is an artifact of the Gaussian amortization.
  • Domain-decomposition or multi-scale extensions suggested by the authors would be the natural next stress test for conservation-by-construction under spectral bias.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a variational latent neural-field (VLNF) framework for parameterized nonlinear conservation laws that combines reduced-order latent dynamics with Gaussian-process-inspired uncertainty quantification. Three variants are developed: IRS-UQ (unconstrained), PI-IRS-UQ (soft physics residual penalty), and ECLEIRS-UQ (exact conservation by construction). Uncertainty is obtained by modeling the latent state as a GP over parameter space, with posterior mean and diagonal variance amortized by neural surrogates (GPMN/GPVN), then decoded by an INR; temporal evolution of the mean is learned via a neural ODE. Exact conservation in ECLEIRS-UQ is enforced by representing the random solution-flux field as the row-wise divergence of a skew-symmetric matrix potential (Eqs. 36–43), so every realization is space-time divergence-free; Theorem 1 shows the mean inherits this property under a mild commutativity assumption. Three parameterized benchmarks (1-D advection, 2-D Euler density, 2-D shallow water) with sparsity/noise ablations and β studies report relative error, conservation residual, error–SD correlation, and 2σ coverage, arguing that ECLEIRS-UQ improves robustness under degraded data while remaining competitive on UQ metrics.

Significance. If the claims hold, the work fills a genuine gap: simultaneous exact conservation-structure preservation and parameter-space UQ in non-intrusive latent dynamics / neural-field ROMs. The construction is clean (skew-symmetric potential ⇒ ∇·z=0 by identity; Theorem 1 is elementary once samples lie on the manifold), the amortized GP surrogates address the usual O(N³) GP bottleneck, and the three graduated experiments with explicit conservation residual, correlation, and coverage metrics under sparsity/noise are stronger evidence than is common in this literature. The framework is also stated to transfer to latent DeepONet-style operators. These are concrete, usable contributions for multi-query scientific computing where both physical consistency and OOD confidence matter.

major comments (3)
  1. §3.1, Eqs. (14)–(20): The load-bearing UQ claim (intrinsic OOD confidence via GP covariance over ν) rests on a diagonal amortized GP posterior (Σ_GP≈σ²_GP I; correlations only across parameters; time only in the mean). The manuscript does not quantify how well GPMN/GPVN approximate the true GP posterior under distribution shift, nor the effect of latent-component coupling. A short ablation (exact vs amortized GP on a small training set, or off-diagonal residual) would make the OOD confidence claim more defensible; without it the UQ story is plausible but not fully stress-tested.
  2. §7 (esp. 7.2–7.3) and abstract claim of competitive UQ: Correlation coefficients for 2-D Euler and shallow water are only moderate (~0.3–0.6), with visibly weaker correlation away from shocks/waves and incomplete coverage near bathymetry. The paper correctly notes this pattern, but the abstract and conclusions still present UQ as broadly successful. The claim should be narrowed to “informative uncertainty concentrated near discontinuities and under parameter extrapolation,” with clearer discussion of when coverage/correlation degrade.
  3. §5.3 and §7.2: ECLEIRS-UQ requires solution and flux data and, for Euler, only enforces continuity. Consistency between the reconstructed flux and the constitutive physical flux is acknowledged as data-dependent but not quantified. A residual between modeled and constitutive flux (or multi-equation conservation residuals) would clarify what “exact conservation” delivers in multi-physics settings and whether the robustness gains survive when only partial conservation is imposed.
minor comments (5)
  1. Notation: σ_n is used both for GP observation noise (§3.1) and for added data noise σ_N in §7; unify or disambiguate.
  2. Table 1 / §6: Latent dimension is set to d_ν+2 with no sensitivity study; a one-sentence justification or small sweep would help reproducibility.
  3. Figure 1 caption and §6.1: Segregated training (GPVN then GPMN+decoder then neural ODE) is clear, but joint vs segregated trade-offs are not discussed; a brief remark would help practitioners.
  4. Typos / polish: author name spacing (“A viral”), occasional missing spaces in math, and “PI-URS-UQ” in §7.3 should be corrected.
  5. Related work: GPLaSDI [25] and other latent GP / Bayesian operator papers are cited; a sharper one-paragraph contrast on what amortization + exact conservation adds would strengthen positioning.

Circularity Check

1 steps flagged

No significant circularity: conservation is intentionally by construction (skew-symmetric potential), UQ training is standard ELBO/reconstruction, and self-citation to prior ECLEIRS supplies the structure-preserving block without forcing the new UQ claims or experimental metrics.

specific steps
  1. self citation load bearing [§1.3, §2.2, §5 (esp. 5.1–5.3) and naming of ECLEIRS-UQ]
    "Recent work [14] proposed a structure-preserving latent-dynamics framework that embeds the nonlinear conservation law directly into the solution representation, thereby guaranteeing exact conservation. ... In [14], this identity was used to impose nonlinear conservation law structure when modeling deterministic solution fields using reduced state dynamics approaches. ... As this approach can be considered as a probabilistic extension of the ECLEIRS framework proposed in [14], we refer to it as ECLEIRS-UQ."

    The exact space-time divergence-free construction that defines the flagship ECLEIRS-UQ variant (skew-symmetric potential AAA whose row-wise divergence yields the solution-flux vector) is imported wholesale from the authors’ own prior deterministic paper [14]. While the underlying vector-calculus/exterior-calculus identities are classical and externally cited, the concrete latent-dynamics realization and the robustness narrative under sparse/noisy data rest on that self-citation. The circularity is limited because the present work’s new content (variational GP-inspired UQ, amortized GPMN/GPVN, three-variant comparison, and all reported error/coverage/correlation metrics) does not reduce to the citation.

full rationale

The derivation chain is self-contained for the novel parts. Latent random fields are defined via amortized GP mean/variance networks plus a decoder (Eqs. 9–20, 23); training is ordinary reconstruction + KL (optionally + residual penalty for PI-IRS-UQ). Exact conservation in ECLEIRS-UQ follows immediately from the classical identity that the row-wise divergence of a skew-symmetric matrix field is divergence-free (Eqs. 32–43, citing exterior-calculus sources [48,49] and the authors’ deterministic ECLEIRS [14]); Theorem 1 is a one-line commutativity argument that the mean inherits the property. Numerical claims (relative error, correlation of SD with error, 2σ coverage, robustness under sparsity/noise) are evaluated on held-out parameter instances against high-fidelity data and are not algebraically forced by the training losses or by the conservation construction. The only mild circularity is ordinary self-citation of [14] for the deterministic structure-preserving decoder; that citation is not used to import a uniqueness theorem or to rename a fitted quantity as a prediction, and the UQ layer plus the three-variant comparison constitute independent content. Score 2 reflects that single non-load-bearing self-citation; the paper does not reduce its central accuracy/UQ claims to its own inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 2 invented entities

The central claims rest on standard PDE/ROM math, a GP-VAE modeling stack with several free regularization and architecture choices, and the authors’ prior exact-conservation decoder idea extended to random fields. No new physical particles or forces are postulated; the ‘invented’ pieces are methodological constructs (VLNF, GPVN/GPMN, ECLEIRS-UQ).

free parameters (6)
  • KL weight β = typically 1e-4 after sweep
    Controls accuracy–coverage tradeoff; selected by violin-plot sweeps (e.g., β=10^{-4} used across experiments).
  • Physics penalty λ (PI-IRS-UQ) = problem-dependent (e.g. 1e-3 or 1e-1)
    Soft conservation weight; results sensitive to λ (10^{-3} vs 10^{-1}) across problems.
  • Latent dimension r = d_ν + 2
    Set by hand to dν+2 for all experiments.
  • GP hyperparameters (σ_f, length scales l, σ_n) = learned per problem
    Jointly optimized with decoder/GPMN in the ELBO objective; define OOD variance behavior.
  • Network widths/depths and SIREN architecture = see Table 1
    Fixed per Table 1; capacity choices affect reconstruction and UQ quality.
  • Monte Carlo sample count for predictive moments = 10
    Ten latent samples used for mean/SD/coverage; affects UQ metric stability.
axioms (7)
  • domain assumption Target systems admit a nonlinear conservation-law form R(q;ν)=∂q/∂t+∇·f(q;ν)=0 (or mass conservation subset for multi-physics).
    Stated in §2.1; ECLEIRS-UQ structure only applies when this form is the constraint of interest.
  • ad hoc to paper Latent state prior is a GP over parameter space with squared-exponential kernel; posterior covariance is diagonalized as σ²_GP I.
    §3.1 Eqs. 11–16; independence of latent components is an explicit scalability assumption.
  • ad hoc to paper Amortized neural surrogates (GPMN, GPVN) adequately replace exact GP posterior mean/variance for training and inference.
    §3.1 Eqs. 17–20; core of claimed scalable OOD UQ.
  • standard math VAE-style Gaussian likelihood + ELBO with reparameterization is a valid training objective for the random solution field.
    §3.2; standard variational inference assumptions.
  • standard math Skew-symmetric matrix potential (or curl in low dimension) yields exact space-time divergence-free solution-flux fields, including for random realizations.
    §5.1–5.3 citing exterior-calculus identities [48,49]; extended to random fields.
  • standard math Expectation and space-time divergence commute under square-integrability/regularity (Theorem 1).
    Needed so the predictive mean remains conservation-law consistent.
  • domain assumption Flux (and solution) data are available for PI-IRS-UQ and ECLEIRS-UQ training.
    Stated in §5–6; limits purely solution-only datasets.
invented entities (2)
  • Variational latent neural field (VLNF) with GPMN/GPVN surrogates no independent evidence
    purpose: Represent parameterized PDE solutions as random fields with amortized GP-like latent uncertainty.
    Methodological construct combining INR auto-decoders, VAE sampling, and GP amortization; not a physical entity.
  • ECLEIRS-UQ (and IRS-UQ / PI-IRS-UQ variants) no independent evidence
    purpose: Name the unconstrained, soft-physics, and exact-conservation UQ latent-dynamics formulations.
    Named method variants; ECLEIRS-UQ extends authors’ prior ECLEIRS to probabilistic fields.

pith-pipeline@v1.1.0-grok45 · 38844 in / 4090 out tokens · 41271 ms · 2026-07-14T08:00:24.536238+00:00 · methodology

0 comments
read the original abstract

Reduced-order models, such as latent dynamics models, are becoming mainstream for accelerating simulations for parameterized physical systems governed by nonlinear conservation laws. However, most existing latent dynamics frameworks suffer from two important limitations: they do not provide uncertainty estimates for model predictions, and they do not guarantee adherence to the underlying conservation laws. While these challenges have been addressed separately in prior work, a unified framework that simultaneously provides uncertainty quantification and exact conservation-law preservation remains largely unexplored. In this work, we develop a variational latent neural field framework that integrates Gaussian process-inspired surrogates, enabling estimation of predictive confidence for both in-distribution and out-of-distribution parameter regimes. Three variants of the framework are considered: IRS-UQ, PI-IRS-UQ, and ECLEIRS-UQ, corresponding to unconstrained, physics-informed, and conservation-structure-preserving formulations, respectively. Exact conservation-structure preservation is achieved by embedding the solution dynamics within a conservation-law manifold through a space-time divergence-free representation of the solution-flux field. We demonstrate the applicability of the framework through three numerical experiments: 1) 1-D advection, 2) 2-D Euler and 3) 2-D shallow water equations in parameterized settings. Numerical experiments demonstrate that the proposed approach provides accurate predictions together with uncertainty estimates, while remaining robust to sparse and noisy training data. Comparisons between the proposed three approaches show that conservation-structure preserving latent representations improve robustness to degraded training data while maintaining competitive predictive accuracy and uncertainty quantification capability.

Figures

Figures reproduced from arXiv: 2607.10965 by Aviral Prakash, Marc L. Klasky.

Figure 1
Figure 1. Figure 1: Neural network architectures for IRS-UQ, PI-IRS-UQ and ECLEIRS-UQ. The dia [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 1-D advection problem: Space-time sensor locations for training different reduced [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 1-D advection problem: Violin plot showing the distribution with respect to the [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 1-D advection problem: Solution prediction by ECLEIRS-UQ trained on full reso [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 1-D advection problem: Space-time contour of solution prediction by ECLEIRS-UQ [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 1-D advection problem: Violin plot showing the distribution of metrics with respect [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: 1-D advection problem: Violin plot showing the distribution with respect to the sys [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The selection of parameter points used in the learning dataset (blue markers) and [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 2-D Euler problem: Spatial locations of data used to learn different reduced state [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 2-D Euler problem: Violin plot showing the distribution with respect to the system [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: 2-D Euler problem: Contour of density prediction at [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: 2-D Euler problem: Violin plot showing the distribution of relative errors and [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: 2-D Euler problem: Violin plot showing the distribution of relative errors and [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: 2-D shallow water problem: Violin plot showing the distribution with respect to the [PITH_FULL_IMAGE:figures/full_fig_p033_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: 2-D shallow water-depth prediction: Contour of water depth prediction at [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: 2-D shallow water problem: Violin plot showing the distribution of relative errors and [PITH_FULL_IMAGE:figures/full_fig_p035_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: 2-D Shallow water problem: Violin plot showing the distribution of relative errors [PITH_FULL_IMAGE:figures/full_fig_p036_17.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

59 extracted references · 10 linked inside Pith

  1. [1]

    Turbulence and the dynamics of coherent structures part III: Dy- namics and scaling

    Lawrence Sirovich. Turbulence and the dynamics of coherent structures part III: Dy- namics and scaling. Quarterly of Applied Mathematics , 45(3):583–590, 1987

    1987

  2. [2]

    Lumley, and Emily Stone

    Nadine Aubry, Philip Holmes, John L. Lumley, and Emily Stone. The dynamics of coherent structures in the wall region of a turbulent boundary layer. Journal of Fluid Mechanics, 192:115173, 1988

    1988

  3. [3]

    A survey of projection-based model reduction methods for parametric dynamical systems

    Peter Benner, Serkan Gugercin, and Karen Willcox. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Review , 57(4):483–531, 2015

    2015

  4. [4]

    Benjamin Peherstorfer and Karen E. Willcox. Data-driven operator inference for non- intrusive projection-based model reduction. Computer Methods in Applied Mechanics and Engineering, 306:196–215, 2016. 37

    2016

  5. [5]

    Boris Kramer, Benjamin Peherstorfer, and Karen E. Willcox. Learning nonlinear re- duced models from data with operator inference. Annual Review of Fluid Mechanics , 56:521–548, 2024

    2024

  6. [6]

    Adjacency-based, non-intrusive model reduction for vortex-induced vibrations

    Leonidas Gkimisis, Thomas Richter, and Peter Benner. Adjacency-based, non-intrusive model reduction for vortex-induced vibrations. Computers & Fluids , 275:106248, 2024

    2024

  7. [7]

    A viral Prakash and Yongjie J. Zhang. Data-driven identification of stable sparse differ- ential operators using constrained regression. Computer Methods in Applied Mechanics and Engineering, 429:117149, 2024

    2024

  8. [8]

    A viral Prakash and Yongjie J. Zhang. Nonintrusive projection-based reduced order modeling using stable learned differential operators. Computer Methods in Applied Mechanics and Engineering , 442:117946, 2025

    2025

  9. [9]

    Fourier Neural Operator for Para- metric Partial Differential Equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhat- tacharya, Andrew Stuart, and Anima Anandkumar. Fourier Neural Operator for Para- metric Partial Differential Equations. arXiv: 2010.08895 , 2021

    Pith/arXiv arXiv 2010

  10. [10]

    Learning nonlinear operators via DeepONet based on the universal approximation the- orem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation the- orem of operators. Nature Machine Intelligence , 3(3):218–229, 2021

    2021

  11. [11]

    Fries, Xiaolong He, and Youngsoo Choi

    William D. Fries, Xiaolong He, and Youngsoo Choi. LaSDI: Parametric latent space dynamics identification. Computer Methods in Applied Mechanics and Engineering , 399:115436, 2022

    2022

  12. [12]

    Kookjin Lee and Eric J. Parish. Parameterized neural ordinary differential equations: Applications to computational physics problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 477, 2021

    2021

  13. [13]

    Park, William Fries, Daniel A

    Christophe Bonneville, Xiaolong He, April Tran, Jun S. Park, William Fries, Daniel A. Messenger, Siu W. Cheung, Yeonjong Shin, David M. Bortz, Debojyoti Ghosh, Jiun- Shyan Chen, Jonathan Belof, and Youngsoo Choi. A comprehensive review of latent space dynamics identification algorithms for intrusive and non-intrusive reduced-order- modeling. arXiv:2403.10748, 2024

    Pith/arXiv arXiv 2024

  14. [14]

    Southworth, and Marc L

    A viral Prakash, Ben S. Southworth, and Marc L. Klasky. Ecleirs: Exact conservation law embedded identification of reduced states for parameterized nonlinear conservation laws from sparse and noisy data. Journal of Computational Physics , 550:114651, 2026

    2026

  15. [15]

    Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators

    Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chat- topadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Aziz- zadenesheli, Pedram Hassanzadeh, Karthik Kashinath, and Animashree Anandkumar. Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators. arXiv:2202...

    Pith/arXiv arXiv 2022

  16. [16]

    Plasma surrogate modelling using fourier neural opera- tors

    Vignesh Gopakumar, Stanislas Pamela, Lorenzo Zanisi, Zongyi Li, Ander Gray, Daniel Brennand, Nitesh Bhatia, Gregory Stathopoulos, Matt Kusner, Marc Peter Deisenroth, and Anima and Anandkumar. Plasma surrogate modelling using fourier neural opera- tors. Nuclear Fusion, 64(5):056025, April 2024

    2024

  17. [17]

    Carlberg

    Kookjin Lee and Kevin T. Carlberg. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. Journal of Computational Physics , 404:108973, 2020

    2020

  18. [18]

    A fast and ac- curate physics-informed neural network reduced order model with shallow masked au- toencoder

    Youngkyu Kim, Youngsoo Choi, David Widemann, and Tarek Zohdi. A fast and ac- curate physics-informed neural network reduced order model with shallow masked au- toencoder. Journal of Computational Physics , 451:110841, 2022

    2022

  19. [19]

    Chen, Jinxu Xiang, Dong H

    Peter Y. Chen, Jinxu Xiang, Dong H. Cho, Yue Chang, G. A. Pershing, Henrique T. Maia, Maurizio M. Chiaramonte, Kevin T. Carlberg, and Eitan Grinspun. CROM: Continuous reduced-order modeling of PDEs using implicit neural representations. In The Eleventh International Conference on Learning Representations , 2023

    2023

  20. [20]

    Kara, and Yongjie J

    Vedant Puri, A viral Prakash, Levent B. Kara, and Yongjie J. Zhang. SNF-ROM: Projection-based nonlinear reduced order modeling with smooth neural fields. Jour- nal of Computational Physics , 532:113957, 2025

    2025

  21. [21]

    Keo Springer, and Kyle T

    Seung Whan Chung, Christopher Miller, Youngsoo Choi, Paul Tranquilli, H. Keo Springer, and Kyle T. Sullivan. Latent space dynamics identification for interface track- ing with application to shock-induced pore collapse. Journal of Computational Physics , 561:114978, 2026

    2026

  22. [22]

    Aslam, and Marc L

    A viral Prakash, Tariq D. Aslam, and Marc L. Klasky. Accelerating burn region modeling in reactive flows using latent space dynamics modeling. In Division of Fluid Dynamics Annual Meeting, American Physical Society , 2025

    2025

  23. [23]

    Reduced-order modeling for parame- terized PDEs via implicit neural representations

    Tianshu Wen, Kookjin Lee, and Youngsoo Choi. Reduced-order modeling for parame- terized PDEs via implicit neural representations. arXiv:2311.16410, 11 2023

    Pith/arXiv arXiv 2023

  24. [24]

    tlasdi: Thermodynamics-informed latent space dynamics identification

    Jun Sur Richard Park, Siu Wun Cheung, Youngsoo Choi, and Yeonjong Shin. tlasdi: Thermodynamics-informed latent space dynamics identification. Computer Methods in Applied Mechanics and Engineering , 429:117144, 2024

    2024

  25. [25]

    Christophe Bonneville, Youngsoo Choi, Debojyoti Ghosh, and Jonathan L. Belof. Gplasdi: Gaussian process-based interpretable latent space dynamics identification through deep autoencoder. Computer Methods in Applied Mechanics and Engineering , 418:116535, 2024

    2024

  26. [26]

    Approximate bayesian neural operators: Uncertainty quantification for para- metric PDEs

    Emilia Magnani, Nicholas Krämer, Runa Eschenhagen, Lorenzo Rosasco, and Philipp Hennig. Approximate bayesian neural operators: Uncertainty quantification for para- metric PDEs. Transactions on Machine Learning Research , 2025. 39

    2025

  27. [27]

    B-deeponet: An enhanced bayesian deeponet for solving noisy parametric pdes using accelerated replica exchange sgld

    Guang Lin, Christian Moya, and Zecheng Zhang. B-deeponet: An enhanced bayesian deeponet for solving noisy parametric pdes using accelerated replica exchange sgld. Journal of Computational Physics , 473:111713, 2023

    2023

  28. [28]

    Randomized prior wavelet neural operator for uncertainty quantification

    Shailesh Garg and Souvik Chakraborty. Randomized prior wavelet neural operator for uncertainty quantification. Probabilistic Engineering Mechanics, 81:103817, 2025

    2025

  29. [29]

    Psaros, and George E

    Zongren Zou, Xuhui Meng, Apostolos F. Psaros, and George E. Karniadakis. Neuraluq: A comprehensive library for uncertainty quantification in neural differential equations and operators. SIAM Review, 66(1):161–190, 2024

    2024

  30. [30]

    Maddix, Shima Alizadeh, Gaurav Gupta, Andrew Stuart, Michael W

    S Chandra Mouli, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, Andrew Stuart, Michael W. Mahoney, and Bernie Wang. Using uncertainty quantification to character- ize and improve out-of-domain learning for PDEs. In Proceedings of the 41st Interna- tional Conference on Machine Learning , volume 235 of Proceedings of Machine Learning Research, pages 36372...

    2024

  31. [31]

    Scalable uncertainty quantification for deep operator networks using randomized priors

    Yibo Yang, Georgios Kissas, and Paris Perdikaris. Scalable uncertainty quantification for deep operator networks using randomized priors. arXiv preprint arXiv:2203.03048 , 2022

    Pith/arXiv arXiv 2022

  32. [32]

    Calibrated uncertainty quantification for operator learning via conformal prediction

    Ziqi Ma, David Pitt, Kamyar Azizzadenesheli, and Anima Anandkumar. Calibrated uncertainty quantification for operator learning via conformal prediction. Transactions on Machine Learning Research , 2024

    2024

  33. [33]

    Conformalized-deeponet: A distribution-free framework for uncertainty quantification in deep operator networks

    Christian Moya, Amirhossein Mollaali, Zecheng Zhang, Lu Lu, and Guang Lin. Conformalized-deeponet: A distribution-free framework for uncertainty quantification in deep operator networks. Physica D: Nonlinear Phenomena , 471:134418, 2025

    2025

  34. [34]

    Probabilistic neural operators for functional uncertainty quantification

    Christopher Bülte, Philipp Scholl, and Gitta Kutyniok. Probabilistic neural operators for functional uncertainty quantification. Transactions on Machine Learning Research , 2025

    2025

  35. [35]

    Neural operator induced gaus- sian process framework for probabilistic solution of parametric partial differential equa- tions

    Sawan Kumar, Rajdip Nayek, and Souvik Chakraborty. Neural operator induced gaus- sian process framework for probabilistic solution of parametric partial differential equa- tions. Computer Methods in Applied Mechanics and Engineering , 431:117265, 2024

    2024

  36. [36]

    Towards gaussian process for operator learning: An uncertainty aware resolution independent operator learning al- gorithm for computational mechanics

    Sawan Kumar, Rajdip Nayek, and Souvik Chakraborty. Towards gaussian process for operator learning: An uncertainty aware resolution independent operator learning al- gorithm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 435:117664, 2025

    2025

  37. [37]

    Operator learning with gaussian processes

    Carlos Mora, Amin Yousefpour, Shirin Hosseinmardi, Houman Owhadi, and Ramin Bostanabad. Operator learning with gaussian processes. Computer Methods in Applied Mechanics and Engineering , 434:117581, 2025

    2025

  38. [38]

    Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Ma- chine Learning. The MIT Press, 11 2005. 40

    2005

  39. [39]

    Nature Communications, 15:10416, 2024

    Conditional neural field latent diffusion model for generating spatiotemporal turbulence. Nature Communications, 15:10416, 2024

    2024

  40. [40]

    Kingma and Max Welling

    Diederik P. Kingma and Max Welling. An introduction to variational autoencoders. Found. Trends Mach. Learn. , 12(4):307392, November 2019

    2019

  41. [41]

    Katiana Kontolati, Somdatta Goswami, George Em Karniadakis, and Michael D. Shields. Learning in latent spaces improves the predictive accuracy of deep neural operators. arXiv:2304.07599, 2023

    Pith/arXiv arXiv 2023

  42. [42]

    Approximation by superpositions of a sigmoidal function

    G Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems , 2:303–314, 1989

    1989

  43. [43]

    Multilayer feedforward net- works are universal approximators

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward net- works are universal approximators. Neural Networks, 2(5):359–366, 1989

    1989

  44. [44]

    Dogyun Park, Sihyeon Kim, Sojin Lee, and Hyunwoo J. Kim. Ddmi: Domain-agnostic latent diffusion models for synthesizing high-quality implicit neural representations. arXiv:2401.12517, 2024

    Pith/arXiv arXiv 2024

  45. [45]

    Ln3diff++: Scalable latent neural fields diffusion for speedy 3d generation

    Yushi Lan, Fangzhou Hong, Shangchen Zhou, Shuai Yang, Xuyi Meng, Yongwei Chen, Zhaoyang Lyu, Bo Dai, Xingang Pan, and Chen Change Loy. Ln3diff++: Scalable latent neural fields diffusion for speedy 3d generation. IEEE Transactions on Pattern Analysis and Machine Intelligence , 48(3):30023017, March 2026

    2026

  46. [46]

    beta-V AE: Learning basic visual concepts with a constrained variational framework

    Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-V AE: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, 2017

    2017

  47. [47]

    Karniadakis

    Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics , 378:686–707, 2019

    2019

  48. [48]

    Representation of divergence-free vector fields

    Cristian Barbarosie. Representation of divergence-free vector fields. Quarterly of applied mathematics, 69(2):309–316, 2011

    2011

  49. [49]

    Kelliher

    James P. Kelliher. Stream functions for divergence-free vector fields. Quarterly of applied mathematics, 79(1):163–174, 2021

    2021

  50. [50]

    M. Spivak. Calculus On Manifolds . CRC Press, 1965

    1965

  51. [51]

    John M. Lee. Introduction to Smooth Manifolds . Springer New York, 2003

    2003

  52. [52]

    Jack Richter-Powell, Yaron Lipman, and Ricky T. Q. Chen. Neural conservation laws: A divergence-free perspective. In Advances in Neural Information Processing Systems , 2022. 41

    2022

  53. [53]

    Vincent Sitzmann, Julien N. P. Martel, Alexander W. Bergman, David B. Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. In Proceedings of the 34th International Conference on Neural Information Processing Systems, NIPS ’20, 2020

    2020

  54. [54]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv:1412.6980, 2017

    Pith/arXiv arXiv 2017

  55. [55]

    Decoupled weight decay regularization

    Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv:1711.05101, 2019

    Pith/arXiv arXiv 2019

  56. [56]

    Efficient implementation of weighted ENO schemes

    Guang-Shan Jiang and Chi-Wang Shu. Efficient implementation of weighted ENO schemes. Journal of Computational Physics , 126(1):202–228, 1996

    1996

  57. [57]

    Efficient implementation of essentially non-oscillatory shock-capturing schemes

    Chi-Wang Shu and Stanley Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics , 77(2):439–471, 1988

    1988

  58. [58]

    Kingma, Abhishek Kumar, Ste- fano Ermon, and Ben Poole

    Yang Song, Jascha Narain Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Ste- fano Ermon, and Ben Poole. Score-based generative modeling through stochastic dif- ferential equations. arXiv:2011.13456, abs/2011.13456, 2020

    Pith/arXiv arXiv 2011

  59. [59]

    Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations , 2023. 42

    2023