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arxiv: 2606.02427 · v1 · pith:K3USXK3Wnew · submitted 2026-06-01 · 🧮 math.NA · cs.LG· cs.NA

Spectral Audit of In-Context Operator Networks

Zhiwei Gao , Liu Yang , George Em Karniadakis This is my paper

Pith reviewed 2026-06-28 13:33 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA
keywords neural operatorsin-context learningJacobian spectral audittangent operatorFourier projectionoperator fidelityPDE learning
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The pith

Neural operators can match solution outputs while learning incorrect local PDE structure.

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

The paper establishes that prediction accuracy alone does not confirm a neural operator has learned the correct local dynamics of the underlying PDE, because a model may reproduce outputs yet exhibit wrong sensitivities, frequency responses, or mode couplings. It proposes a Jacobian-based spectral audit that differentiates the network output with respect to the query function for a fixed prompt, treats the result as a learned tangent operator, and projects it onto Fourier modes to extract gains, phases, and cross-mode terms. This audit exposes operator-level phenomena such as phase transport, viscosity-dependent damping, and nonlinear coupling that remain invisible to standard error metrics, and it flags cases where corrupted prompts degrade the tangent structure even when pointwise predictions stay partially accurate. The distinction matters for any downstream use that depends on faithful reproduction of the governing mechanisms rather than isolated solution matches.

Core claim

Differentiating an in-context operator network output with respect to the query function produces a Jacobian that, when projected onto Fourier modes, yields a local spectral characterization of the inferred tangent operator, including frequency-dependent gains, phase structure, and cross-mode coupling. Across benchmarks the resulting audit detects phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction-diffusion stability features while also identifying high-frequency degradation, incorrect phase recovery, and prompt-operator inconsistencies that prediction-error metrics leave partially hidden.

What carries the argument

The Jacobian of the network output with respect to the query function, interpreted as a learned tangent operator and projected onto Fourier modes to obtain frequency gains, phases, and coupling coefficients.

If this is right

  • Prediction error alone cannot certify that a learned operator reproduces the correct local PDE mechanisms.
  • Corrupted or inconsistent prompts produce degraded tangent-operator structure even when pointwise predictions remain partially accurate.
  • The audit can detect high-frequency degradation and incorrect phase recovery that standard metrics miss.
  • Operator fidelity must be assessed separately from solution accuracy whenever stability or sensitivity information is required.

Where Pith is reading between the lines

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

  • Training objectives for in-context operator networks could incorporate spectral penalties on the Jacobian to enforce local fidelity in addition to output matching.
  • The audit extends naturally to time-evolving or multi-physics operators where local linearization can reveal stability boundaries not visible in static error plots.
  • Prompt construction in in-context learning may need to optimize for consistency of the inferred tangent operator rather than example accuracy alone.

Load-bearing premise

The finite-difference or automatic-differentiation Jacobian computed from the network output accurately represents the tangent operator of the true PDE at the operating point set by the prompt.

What would settle it

On a linear PDE with known exact Fourier multiplier, compute the audited Jacobian spectrum from the trained network and check whether it systematically deviates from the true multiplier while prediction error on solution outputs remains low.

Figures

Figures reproduced from arXiv: 2606.02427 by George Em Karniadakis, Liu Yang, Zhiwei Gao.

Figure 1
Figure 1. Figure 1: Advection relative L 2 prediction error over hidden speed. The shaded region denotes the training range. Because advection is linear, the query Jacobian should be exactly the same shift operator for every base state: Du0G T c (u0)v(x) = v((x − cT) mod 1). 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Advection query spectrum over hidden speed. The top row shows ICON Fourier-mode gains, and the bottom row reports accumulated relative spectral errors. For transport, however, gain alone is not enough. A map may preserve the norm of each Fourier mode while rotating the sine–cosine pair by the wrong angle, which would correspond to an incorrect shift. This is why the phase/block error in [PITH_FULL_IMAGE:f… view at source ↗
Figure 3
Figure 3. Figure 3: Advection phase and block-rotation error over hidden speed [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fourier feature maps for advection at c = 0.80 (shift s = 0.40), using the first 16 sine–cosine Fourier pairs: exact block-rotation map, ICON Jacobian map, and normalized entrywise error map. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Burgers relative L 2 prediction error with respect to the true finite-time solution GT c u0. The shaded region denotes the training range. We next audit the context-conditioned query Jacobian at controlled base states. For a query trajectory u(t, x), the true tangent perturbation satisfies vt + uvx + vux = cvxx, v(0, x) = v0(x). The context controls the hidden viscosity c, which determines the diffusive te… view at source ↗
Figure 6
Figure 6. Figure 6: Burgers context-conditioned query Jacobian spectra at deterministic base states. Each row fixes one query base state, and different curves correspond to different context viscosities. The Fourier mode k denotes the perturbation direction applied to the query input. Gain curves show context-conditioned damping, while error panels report accumulated relative spectral errors with respect to the true Burgers t… view at source ↗
Figure 7
Figure 7. Figure 7: Fourier feature maps for Burgers at c = 0.02, evaluated around the smooth, steep-gradient, and near-shock-like base states and projected onto the first 14 sine–cosine Fourier pairs. True and ICON maps use a common color scale; error maps show normalized entrywise discrepancy between the learned and true Fourier-projected Jacobians. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Clean Burgers evaluation of the corrupted-prompt model. The model is trained with inconsistent context–query viscosities, but the test prompts use a single correct viscosity c. The prediction error in [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Query-spectrum diagnostic for the corrupted-prompt Burgers model evaluated on clean prompts. The true reference is the clean Burgers tangent spectrum, and the error panels report accumulated relative spectral error [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fourier feature map for the corrupted-prompt Burgers model. The true map is computed from the clean Burgers tangent equation, while the ICON map is obtained from the model trained on inconsistent prompts. The error map visualizes the resulting operator-structure mismatch. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Allen–Cahn relative L 2 prediction error with respect to the true finite-time solution. The shaded region denotes the training range. We next audit the context-conditioned query Jacobian at controlled base states. For a trajectory u(t, x), the true tangent perturbation satisfies vt = ε 2 vxx + (1 − 3u(t, x) 2 )v, v(0, x) = v0(x). The role of the hidden parameter is seen from the diffusion part of this tan… view at source ↗
Figure 12
Figure 12. Figure 12: Allen–Cahn context-conditioned query Jacobian spectra at deterministic base states. Each row fixes one query base state, and different curves correspond to different context values of ε. The Fourier mode k denotes the perturbation direction applied to the query input. Gain curves show the learned reaction–diffusion response, while error panels report accumulated relative spectral errors with respect to th… view at source ↗
Figure 13
Figure 13. Figure 13: Fourier feature maps for Allen–Cahn at ε = 0.05, evaluated around the near-zero, moderate￾amplitude, and interface-like base states and projected onto the first 16 sine–cosine Fourier pairs. True and ICON maps use a common color scale; error maps show normalized entrywise discrepancy between the learned and true Fourier-projected Jacobians. The Allen–Cahn audit therefore tests a local stability mechanism … view at source ↗
Figure 14
Figure 14. Figure 14: Burgers predictions for the smooth, steep-gradient, and near-shock-like base states. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Allen–Cahn predictions for the near-zero, moderate-amplitude, and interface-like base states [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Advection bandwidth ablation. The two models are trained with ntrain = 8 and ntrain = 16, respectively, and are evaluated on test functions with bandwidth neval = 8 and neval = 16. The shaded region denotes the training range for the hidden speed c. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Advection bandwidth ablation at c = 0.80. The plot compares the query Fourier spectra and accumulated relative spectral errors of the models trained with 8 and 16 Fourier modes. The query-spectrum ablation in [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Phase and block-rotation errors for the advection bandwidth ablation at c = 0.80. Because the true advection Jacobian is a direct sum of 2 × 2 Fourier rotation blocks, this diagnostic directly measures the learned shift phase [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Fourier feature-map ablation for advection at c = 0.80. The exact shift map is compared with ICON models trained on 8 and 16 Fourier modes. The projection uses the first 16 sine–cosine Fourier pairs, so modes beyond k = 8 test frequency extrapolation for the 8-mode model. C True Tangent Maps for the Reference Operators This appendix derives the reference tangent maps used in the spectral audit. For each P… view at source ↗
read the original abstract

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces a Jacobian-based spectral audit for in-context operator networks. For a fixed prompt, the network output is differentiated with respect to the query function to obtain a Jacobian viewed as a learned tangent operator; this is projected onto Fourier modes to yield a local spectral characterization (frequency-dependent gains, phase, cross-mode coupling). The audit is positioned as complementary to prediction-error metrics, revealing phenomena such as phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction-diffusion stability, while detecting failures (high-frequency degradation, incorrect phase recovery, prompt-operator inconsistencies) that prediction accuracy alone misses. The central conclusion is that prediction accuracy and local operator fidelity are distinct properties of learned neural operators.

Significance. If the Jacobian obtained via finite differences or automatic differentiation accurately recovers the Fréchet derivative of the underlying PDE solution operator, the spectral audit supplies a new diagnostic that can expose structural mismatches invisible to pointwise error metrics. This would be a useful addition to evaluation practices for neural operators in scientific computing, particularly for assessing stability, sensitivity, and consistency in in-context settings. The framework is parameter-free in its core construction and directly falsifiable via comparison against known linearizations of benchmark PDEs.

major comments (2)
  1. [Abstract] The central claim that the audit detects incorrect local dynamical structure (phase transport, mode coupling, stability) rests on the unverified assumption that the network Jacobian coincides with the tangent operator of the fixed PDE at the prompt-defined operating point rather than an artifact of prompt construction, discretization, or the network's own approximation error. No quantitative verification, error bars, or ablation on the Jacobian approximation itself is supplied.
  2. [Abstract] The distinction between prediction accuracy and local operator fidelity is asserted on the basis of benchmark results, yet the manuscript supplies no concrete quantitative evidence (e.g., tables of prediction error versus spectral-audit metrics, or cases where one is good and the other poor) to substantiate that the two properties are in fact separable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation for major revision. We address each major comment below, agreeing that additional quantitative support will strengthen the claims. Revisions will be incorporated in the next version.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the audit detects incorrect local dynamical structure (phase transport, mode coupling, stability) rests on the unverified assumption that the network Jacobian coincides with the tangent operator of the fixed PDE at the prompt-defined operating point rather than an artifact of prompt construction, discretization, or the network's own approximation error. No quantitative verification, error bars, or ablation on the Jacobian approximation itself is supplied.

    Authors: We agree that explicit verification of the Jacobian approximation is needed to confirm it recovers the Fréchet derivative rather than artifacts. In the revised manuscript we will add a dedicated verification subsection for linear benchmark PDEs (e.g., heat and wave equations) where the exact tangent operator is known analytically. This will include direct comparisons of the computed Jacobian against the analytic linearization, with L2 error norms, error bars over multiple discretizations, and ablations on finite-difference step size and prompt length. These additions will quantify the fidelity of the Jacobian step and rule out the listed artifacts for the reported cases. revision: yes

  2. Referee: [Abstract] The distinction between prediction accuracy and local operator fidelity is asserted on the basis of benchmark results, yet the manuscript supplies no concrete quantitative evidence (e.g., tables of prediction error versus spectral-audit metrics, or cases where one is good and the other poor) to substantiate that the two properties are in fact separable.

    Authors: The current manuscript illustrates separability through qualitative examples (e.g., low prediction error yet incorrect phase recovery or high-frequency degradation). To provide the requested concrete evidence we will add a summary table in the results section that reports, for each model-prompt pair, both relative L2 prediction error and key spectral-audit quantities (maximum gain deviation, phase error at dominant frequencies, and cross-mode coupling strength). Rows will highlight instances where prediction error remains below 5% while spectral metrics exceed acceptable thresholds, thereby quantifying the claimed distinction. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the spectral audit definition or claims

full rationale

The paper introduces a Jacobian-based spectral audit as a new diagnostic tool defined directly by differentiating the network output w.r.t. the query function and projecting onto Fourier modes. This construction is presented as a method to complement prediction error, with no load-bearing step that reduces a claimed prediction or first-principles result to a fitted parameter, self-citation chain, or definitional tautology. The distinction between prediction accuracy and operator fidelity is demonstrated empirically across benchmarks rather than derived by construction from the audit inputs themselves. No self-definitional, fitted-input, or uniqueness-imported patterns appear in the provided abstract or described framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the assumption that automatic differentiation yields a faithful tangent operator and that Fourier projection is the appropriate basis for local spectral analysis; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Jacobian of the network output with respect to the query function equals the tangent operator of the underlying PDE at the prompt point.
    Invoked when the audit treats the computed Jacobian as the learned tangent operator.
  • domain assumption Fourier-mode projection extracts the relevant frequency-dependent gains, phase, and cross-mode coupling.
    Used to obtain the local spectral characterization.

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discussion (0)

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Forward citations

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Reference graph

Works this paper leans on

24 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218–229, 2021

    2021

  2. [2]

    Neural operator: Learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    2023

  3. [3]

    Laplace neural operator for solving differential equations.Nature Machine Intelligence, 6(6):631–640, 2024

    Qianying Cao, Somdatta Goswami, and George Em Karniadakis. Laplace neural operator for solving differential equations.Nature Machine Intelligence, 6(6):631–640, 2024

    2024

  4. [4]

    Wavelet neural operator: a neural operator for parametric partial differential equations.arXiv preprint arXiv:2205.02191, 2022

    Tapas Tripura and Souvik Chakraborty. Wavelet neural operator: a neural operator for parametric partial differential equations.arXiv preprint arXiv:2205.02191, 2022

    arXiv 2022

  5. [5]

    Zhiping Mao, Lu Lu, Olaf Marxen, Tamer A Zaki, and George Em Karniadakis. Deepm&mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators.Journal of computational physics, 447:110698, 2021

    2021

  6. [6]

    Kamaljyoti Nath, Varun Kumar, Daniel J Smith, and George Em Karniadakis. A digital twin for diesel engines: Operator-infused physics-informed neural networks with transfer learning for engine health monitoring.Engineering Applications of Artificial Intelligence, 170:114052, 2026

    2026

  7. [7]

    A foundation model for the earth system.Nature, 641(8065):1180–1187, 2025

    Cristian Bodnar, Wessel P Bruinsma, Ana Lucic, Megan Stanley, Anna Allen, Johannes Brandstetter, Patrick Garvan, Maik Riechert, Jonathan A Weyn, Haiyu Dong, et al. A foundation model for the earth system.Nature, 641(8065):1180–1187, 2025

    2025

  8. [8]

    In-context operator learning with data prompts for differential equation problems.Proceedings of the National Academy of Sciences, 120(39):e2310142120, 2023

    Liu Yang, Siting Liu, Tingwei Meng, and Stanley J Osher. In-context operator learning with data prompts for differential equation problems.Proceedings of the National Academy of Sciences, 120(39):e2310142120, 2023

    2023

  9. [9]

    Pde generalization of in-context operator networks: A study on 1d scalar nonlinear conservation laws.Journal of Computational Physics, 519:113379, 2024

    Liu Yang and Stanley J Osher. Pde generalization of in-context operator networks: A study on 1d scalar nonlinear conservation laws.Journal of Computational Physics, 519:113379, 2024

    2024

  10. [10]

    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

  11. [11]

    Dgm: A deep learning algorithm for solving partial differential equations.Journal of computational physics, 375:1339–1364, 2018

    Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algorithm for solving partial differential equations.Journal of computational physics, 375:1339–1364, 2018

    2018

  12. [12]

    Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    2021

  13. [13]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the national academy of sciences, 113(15):3932–3937, 2016

    Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the national academy of sciences, 113(15):3932–3937, 2016

    2016

  14. [14]

    Data-driven discovery of partial differential equations.Science advances, 3(4):e1602614, 2017

    Samuel H Rudy, Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Data-driven discovery of partial differential equations.Science advances, 3(4):e1602614, 2017. 22

    2017

  15. [15]

    Sensitivity-constrained fourier neural operators for forward and inverse problems in parametric differential equations.arXiv preprint arXiv:2505.08740, 2025

    Abdolmehdi Behroozi, Chaopeng Shen, and Daniel Kifer. Sensitivity-constrained fourier neural operators for forward and inverse problems in parametric differential equations.arXiv preprint arXiv:2505.08740, 2025

    arXiv 2025

  16. [16]

    Forcing and diagnosing failure modes of fourier neural operators across diverse pde families.arXiv preprint arXiv:2601.11428, 2026

    Lennon Shikhman. Forcing and diagnosing failure modes of fourier neural operators across diverse pde families.arXiv preprint arXiv:2601.11428, 2026

    Pith/arXiv arXiv 2026

  17. [17]

    Attention is all you need.Advances in neural information processing systems, 30, 2017

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need.Advances in neural information processing systems, 30, 2017

    2017

  18. [18]

    Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101, 2017

    Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101, 2017

    Pith/arXiv arXiv 2017

  19. [19]

    On the spectral bias of neural networks

    Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. InInternational conference on machine learning, pages 5301–5310. PMLR, 2019

    2019

  20. [20]

    Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines.arXiv preprint arXiv:2602.19265, 2026

    Siavash Khodakarami, Vivek Oommen, Nazanin Ahmadi Daryakenari, Maxim Beekenkamp, and George Em Karniadakis. Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines.arXiv preprint arXiv:2602.19265, 2026

    arXiv 2026

  21. [21]

    A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening.Acta metallurgica, 27(6):1085–1095, 1979

    Samuel M Allen and John W Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening.Acta metallurgica, 27(6):1085–1095, 1979

    1979

  22. [22]

    Free energy of a nonuniform system

    John W Cahn and John E Hilliard. Free energy of a nonuniform system. i. interfacial free energy.The Journal of chemical physics, 28(2):258–267, 1958

    1958

  23. [23]

    Conditional neural processes

    Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and SM Ali Eslami. Conditional neural processes. InInternational conference on machine learning, pages 1704–1713. PMLR, 2018

    2018

  24. [24]

    Neural ordinary differential equations.Advances in neural information processing systems, 31, 2018

    Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations.Advances in neural information processing systems, 31, 2018. 23

    2018