Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

A Physics-preserved Transfer Learning Method for Differential Equations

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read A physics-preserving optimal tensor transport map transfers neural-network solvers of differential equations across domains while keeping the equations' physical relations intact.

desk verdict POTT is a genuinely new transfer method for neural operators with strong empirical gains, but the physics-preservation claim rests on a source-operator regularizer and a proof step that do not hold under operator bias. read the letter →

arxiv 2505.01281 v2 pith:6WTATH7F submitted 2025-05-02 cs.LG cs.AI

Hao-Ran Yang , Chuan-Xian Ren This is my paper · ORCID
classification cs.LGcs.AI
keywords transferlearningneuraloperatordifferentialequationsoptimaltransportphysicspreservationdomainadaptationbiasdistribution
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Domain shift in data-driven differential-equation solvers has two sources: the input-output data distribution changes, and the operator that maps inputs to solutions changes. Existing fixes either retrain on target data (general but data-hungry) or align feature distributions (general but can discard physical structure), while analytic transfer methods preserve physics but apply only to special equations. The proposed POTT (Physics-preserved Optimal Tensor Transport) method learns a physics-regularized optimal transport map between the source and target product distributions $P_s(k,u)$ and $P_t(k,u)$, pushes source samples through the map to form a pseudo-target distribution, and then fine-tunes the pretrained operator on real and transported target samples. The paper reports lower relative error than finetuning and domain-adaptation baselines on Burgers', advection, and Darcy flow tasks, including a roughly 25 percent and 36 percent relative-error reduction on the hardest Darcy task with 50 and 100 target samples, and improved cross-region climate forecasts. If correct, POTT gives neural operators a general transfer procedure that preserves the equations' physical structure when only a small target sample is available.

What carries the argument

The central object is the physics-preserved optimal tensor transport map $T_\theta=(T_k,T_u)$, a neural network that carries each source pair $(k^s,u^s)$ into a transported pair $(k^r,u^r)$; its pushforward $P_r=T_\#P_s$ is meant to approximate the target domain. The map is trained with a min-max dual objective whose Lagrange multiplier $f_\phi$ enforces the marginal constraint, while the physical regularizer $R(T)$ supplies the physics: a problem-specific prior when one exists, otherwise the general consistency term $m(\hat{G}_s(k^r),u^r)$ with the source-trained operator standing in for the true operator. This regularizer is what pulls the transported distribution toward physical input-output relations rather than arbitrary aligned features.

What would settle it

On a shift driven by equation parameters rather than data distribution (for example, Burgers' equation with $\nu=0.01$ in the source and $\nu=0.1$ in the target), compute the PDE residual of POTT's transported samples under both equations; if the transported samples satisfy the source equation better than the target equation, the general regularizer is enforcing the wrong physics, contradicting the physics-preservation claim for operator-bias shifts.

Watch

Extended reading notes

Core claim

The paper's central claim is that the transfer problem for learned differential-equation solvers is best attacked by transporting the source product distribution $P_s(k,u)$ to the target $P_t(k,u)$ with a physics-constrained map $T=(T_k,T_u)$, rather than by aligning abstract features or fine-tuning directly. Formally, POTT solves $$\inf_{T_\#P_s=P_t} \int c((k,u),T(k,u))\,dP_s + R(T),$$ where the regularizer $R$ encodes physics: a known prior when one is available, and in the general case an operator-consistency term $R(T)=m(\hat{G}_s T_k(k^s), T_u(u^s))$ that asks the transported input-output pairs to be consistent with the source-trained operator $\hat{G}_s$. The map is obtained as a saddle point of the unconstrained dual problem, and the consistency theorem shows the saddle point equals the physics-regularized Kantorovich problem, degenerating to a Monge map if the pushforward matches the target exactly. The transported samples $D_r$ then augment the scarce real target samples $D_t$ for fine-tuning, which is how POTT claims to correct operator bias while preserving target physical relations.

Load-bearing premise

The load-bearing premise is that the source-trained operator used in the general physics regularizer is a trustworthy stand-in for the target equation's physics; if the target differs by changed equation parameters, the regularizer can enforce source physics on transported samples instead of preserving target physics.

Editorial extensions

If this is right

  • One transfer pipeline can serve different neural-operator backbones and equation families, since POTT is agnostic to the architecture and needs only source/target samples plus a regularizer.
  • With tens of target samples, POTT can cut relative error by roughly a quarter to a third compared with finetuning on severe domain shifts, moving the transferred model closer to the performance of a model trained with full target data.
  • The pushforward distribution augments the small target set with physically plausible samples, so fewer real target samples are needed to correct operator bias.
  • Physics regularization preserves structures that pure distribution alignment loses, so transferred predictions remain interpretable as solutions of the equation rather than as aligned features.
  • On real-world climate data, the method narrows the gap between a source-region model and a model trained with full target data, supporting use outside synthetic benchmarks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the general regularizer in Eq. (12) is safest when source and target operators are close; under strong operator bias, substituting a target-calibrated operator or a PDE-residual term for $\hat{G}_s$ would test whether physics preservation is genuinely achieved.
  • Editorial inference: because the map decomposes into $T_k$ and $T_u$, POTT could be extended to coupled or multi-scale physical systems by adding separate regularizers per physical quantity, a direction the paper does not explore.
  • Editorial inference: the pushforward distribution could also drive an active-learning loop that selects the real target samples where the model and the transported samples disagree most, potentially extending POTT into the very-small-sample regime the paper identifies as a limitation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes POTT (Physics-preserved Optimal Tensor Transport), a transfer learning method for neural operators solving differential equations. The authors characterize domain shift as a combination of distribution bias and operator bias, learn an optimal transport map between the source and target product distributions, and use the pushforward distribution to generate additional training data for the target domain. The method adds a physics regularization term to the transport objective; when no explicit physical prior is available, the regularization uses the pretrained source operator as a stand-in for the unknown target operator. The paper provides a dual optimization formulation, a consistency theorem (Thm. 4.2), and experiments on Burgers', advection, Darcy flow, and ERA5 climate forecasting tasks, reporting improvements over finetuning and domain adaptation baselines.

Significance. If the theoretical claims and the physics-preservation argument were sound, POTT would be a useful contribution: it targets a real problem (transfer of neural operators under distribution and operator bias), covers several PDE benchmarks and a real-world climate task, reports repeated trials with standard deviations, and shows consistent empirical gains over several strong baselines. However, the paper's central theoretical result (Thm. 4.2) relies on an invalid treatment of the regularization term, and the stated physics-preservation property is not established for the general no-prior case because the regularizer substitutes the source operator for the target operator. These issues are load-bearing for the paper's main claims and require substantive revision. The empirical contribution is still substantial and, in my view, worth pursuing after the theory and the interpretation of the regularization are corrected.

major comments (4)
  1. [Appendix B.1, Eqs. (21)-(22)] The derivation of the dual problem treats the global functional R(T) as a pointwise integrand when forming the c-transform. In Eq. (21), the infimum over T is exchanged with a pointwise supremum over (ξ^r, ζ^r) in Eq. (22), which requires R(T) to be of the form ∫ r(T(x), x)dP_s(x). However, R(T) in Eq. (12) is a functional of the entire transported sample, and R(T) in Eq. (11) is a variance; neither is pointwise in T(x). The 'physics-regularized Kantorovich problem' in Eq. (23) is therefore not derived, and the first assertion of Thm. 4.2 is unsupported. The authors should either restrict R to an additive/pointwise form and prove the required inf-sup exchange under explicit regularity conditions, or provide a different argument that handles the non-additive case.
  2. [Section 4.1, Eq. (12)] Substituting the pretrained source operator Ĝ_s for the unknown operator G makes R(T) = m(Ĝ_s T_k(k^s), T_u(u^s)) a source-physics prior rather than a target-physics regularizer. In the operator-bias setting that motivates the paper (e.g., different viscosity ν in Burgers, Table 6, or different domain geometry in Darcy, Table 8), minimizing this term drives transported pairs toward the source relation u^r = Ĝ_s(k^r), not the target relation u^t = G_t(k^t). These transported pairs are then used to train the target model in the second objective of Eq. (14), so the adapted model is fitted with data generated from source physics. The claim that POTT 'preserves physics' of the target problem is therefore not established for the general no-prior setting. The authors need to either replace Ĝ_s with a well-defined target residual or explicitly limit the claim to cases where the source model is a valid prior for the target operator.
  3. [Appendix B.2, proof of Thm. 4.2] The proof asserts that if T∗#P_s = P_t then R(T∗) = 0 'from the construction of R(T)'. This is false for Eq. (12) when Ĝ_s differs from G_t: under the pushforward distribution, R(T) becomes E_{P_r}[m(Ĝ_s(k), u)], which is a nonzero source-model misfit whenever the source operator does not perfectly match the target operator. It is also not demonstrated for Eq. (11). Since the second bullet of Thm. 4.2 relies on dropping the regularization to identify T∗ with a Monge map, that conclusion is currently unsupported.
  4. [Section 5, Fig. 5 (ablation)] The ablation attributes the preservation of the triangular structure in Darcy D1→D2 to the physical regularization, but the comparison is confounded. The transported source samples (k^r, u^r) enter the task loss in Eq. (14) as well as the regularization term, and the map T itself is trained to match the target marginals. Removing R(T) changes the objective while keeping the same transported samples, so the difference between OTT and POTT in Fig. 5 cannot be uniquely attributed to R(T). A controlled study (e.g., matching the task-loss weights, or isolating the regularizer while keeping the transport objective fixed) is needed to support the physics-preservation claim.
minor comments (4)
  1. [Section 4.2 / Thm. 4.2] The theorem statement writes L(P_s, T∗, f∗) = K(P_s, P_t) + R(T∗), but Eq. (13) contains λR(T); the statement should consistently include the factor λ or explicitly absorb it into R.
  2. [Table 3] The header 'ADVERGE' is a typo and should be 'AVERAGE'.
  3. [Table 12] The z-variable Oracle ClimODE value at 18 hours (12.7 ± 14.4) appears implausibly low compared with the 6-hour (102.9 ± 9.3), 12-hour (134.8 ± 162.3), and 24-hour (193.4 ± 16.3) entries; please check this value.
  4. [Throughout] There are several minor language issues, e.g., 'modelize' (Section 1), 'Comparision' (Fig. 3 caption), 'quite closed to' (Section 5), and 'vertice' (Appendix C.1.2); a careful proofread is recommended.

Circularity Check

2 steps flagged · score 6.0 of 10

The physics-preservation claim reduces to the source operator used inside the regularizer: Eq. (12) defines R through G_hat_s, and Eq. (14) trains the target model on transported pairs whose labels inherit that same source relation.

  1. self definitional [Section 4.1, Eq. (12), and Section 4.2, Theorem 4.2]
    "For the general cases with no physical priors available, we formulate R as the physical relation between the marginal distributions of Pr: R(T) = R(Tk,Tu) = m(G(kr),ur) = m(GTk(ks),Tu(us)) ... In practice, the operator G can be substituted by Ĝs as an approximation."

    The regularizer that is supposed to enforce target-physics preservation is evaluated with the source operator Ĝs. Under the paper's own operator-bias formulation (Eq. 5), Gs != Gt, so minimizing R drives transported pairs toward u^r = Ĝs(k^r), i.e., source physics, not target physics. Consequently, the claim that POTT 'preserves physical information' is not an independent result about the target equation; it is built into the definition of R by replacing the unknown true operator with the source model. Any conclusion that the transported distribution inherits target physics is therefore true only by construction and only if the source operator already matches the target operator.

  2. other [Appendix B.2, proof of Theorem 4.2]
    "With condition T∗#Ps = Pt, the pushforward distribution Pr = Pt, then from the construction of R(T), we have R(T) = 0."

    This is the load-bearing inference of the consistency theorem: it asserts that perfect transport makes the physical regularization vanish, from which the proof concludes that T* is a Monge map and that Pr automatically admits the physics of Pt. But with G replaced by Ĝs, R(T)=0 means Ĝs(k^t)=u^t on target-distributed pairs, which is not implied by Pr=Pt unless the source operator equals the target operator. The proof therefore assumes the very operator-bias correction that POTT is supposed to establish, making the claimed physics-preservation guarantee circular under the implemented approximation.

full rationale

The numerical comparisons to finetuning and DA baselines on held-out target test data are external and are not circular: those error reductions are empirical outcomes, not consequences of the regularizer by construction. However, the paper's central physics-preservation claim and its supporting consistency theorem do reduce to the choice of R in Eq. (12). Because the general regularizer substitutes the pretrained source operator Ĝs for the unknown G, the 'physical relation' enforced on transported samples is the source operator's relation, and Eq. (14) then trains the target model on those transported pairs whose labels inherit that same source relation. Under the paper's own definition of operator bias, this is not target physics; it is an input assumption about the source model. The ablation in Fig. 5 attributes triangular-structure preservation to this regularizer, but the transported samples also enter the task loss and the transported labels are generated through the source-regularized map, so the attribution is confounded. Overall, the empirical results remain independent evidence, but the physics-preservation derivation is partially circular: the preserved physics is defined by the source operator rather than derived from the target problem.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method relies on standard OT dual theory, the existence of a Monge map, and the questionable use of the source model as a physics prior. Key hyperparameters lambda, beta, and training schedules are not reported, and the c-transform derivation assumes R is pointwise when it is not.

free parameters (4)
  • lambda (physics regularization weight) = not reported
    Hyperparameter weighting R(T) in Eq. (13)-(14); no values are given in the paper.
  • beta (weight for transferred source loss) = not reported
    Hyperparameter in Eq. (14) second objective; no values are given in the paper.
  • Training steps N11, N12, N2 = N12 typically 10; others unspecified
    Algorithm 1 specifies the loop structure but not the concrete numbers used in experiments.
  • Cost function c(.,.) = not specified
    The OT cost is not explicitly stated for experiments, likely L2 distance, but this is a design choice that affects the map.
assumptions (4)
  • domain assumption The true operator G is the same across source and target domains, or the source approximation G_hat_s is a good proxy for target physics.
    Required for Eq. (12) to preserve target physics; invalid when the equation changes across domains, for example varying viscosity in Burgers.
  • standard math The dual problem Eq. (13) admits a saddle point and neural min-max training reaches it.
    Invoked in Thm. 4.2 and in the alternating update algorithm; convergence is not verified in practice.
  • domain assumption The Monge map between source and target distributions exists and is approximable by a neural network.
    POTT learns a deterministic T_theta; existence and approximability are not guaranteed for arbitrary continuous distributions.
  • ad hoc to paper The physics regularization R(T) can be pulled inside the pointwise supremum in the c-transform derivation.
    Appendix B.1 Eq. (21)-(22) moves R(T), a global functional of T, inside the supremum over (xi, zeta), which is not mathematically justified.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Physics-preserved Transfer Learning Method for Differential Equations." pith.science (2026). https://pith.science/paper/6WTATH7F

@misc{pith2026250501281,
  author       = {Pith},
  title        = {Pith review of: A Physics-preserved Transfer Learning Method for Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WTATH7F}},
  note         = {Machine review of arXiv:2505.01281}
}
read the original abstract

While data-driven methods such as neural operator have achieved great success in solving differential equations (DEs), they suffer from domain shift problems caused by different learning environments (with data bias or equation changes), which can be alleviated by transfer learning (TL). However, existing TL methods adopted in DEs problems lack either generalizability in general DEs problems or physics preservation during training. In this work, we focus on a general transfer learning method that adaptively correct the domain shift and preserve physical information. Mathematically, we characterize the data domain as product distribution and the essential problems as distribution bias and operator bias. A Physics-preserved Optimal Tensor Transport (POTT) method that simultaneously admits generalizability to common DEs and physics preservation of specific problem is proposed to adapt the data-driven model to target domain utilizing the push-forward distribution induced by the POTT map. Extensive experiments demonstrate the superior performance, generalizability and physics preservation of the proposed POTT method.

Figures

Figures reproduced from arXiv: 2505.01281 by the authors.

Figure 1
Figure 1. Visualization of DA method (COD) in task D3 → D2 on Darcy flow. The 1st and 2nd columns present the input k(x) and output u(x) sample pairs of D3 and D2. The 3rd column vi￾sualizes their feature maps from the aligned dis￾tributions, which are forced to be analogous but lacks clear structure explicit physical meaning. It is unclear whether they retain the correct physical information. The prediction shown in the 4th … view at source ↗
Figure 2
Figure 2. Illustration of POTT. Left: Illustration of problem formulation. The distribution bias leads to the bias of physics, i.e. the operator bias. The goal is to correct the operator bias. Right: POTT correct the operator bias by characterizing Dt in a physics-preserved way. (0) Before the model transfer process, source model Gˆs is pretrained with sufficient source data. (1) The POTT map Tθ between source and target doma… view at source ↗
Figure 3
Figure 3. Comparision with (1) Source-only, source pretrained model, (2) Finetuning on target [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Visualization of Gˆ t on Darcy Flow. The value at each image pixel represents the function value at the sampling point. Brighter colors (yellow) indicate higher values. Columns 1-2 show the input-output function pairs from the target domain. Columns 3-5 show the output…
Figure 5
Figure 5. Figure 5: Visualization of POTT and OTT. Ablation analysis of physical regularization. To investigate the effect of the physical regular￾ization in Eq. (10), we implement an ablation analysis on task D1 → D2 of Darcy flow. As shown in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. MoNo: Multiscale Optimal Transport Neural Operator for Solving PDEs on General Geometries

    cs.LG 2026-08 conditional novelty 5.0 of 10

    MoNo uses entropy-regularized optimal transport to build balanced, stable latent-space projections in a multiscale neural operator, achieving lower relative L2 errors and GFLOPs than prior neural operators.

Reference graph

Works this paper leans on

40 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    Neural optimal transport with general cost functionals

    Arip Asadulaev, Alexander Korotin, Vage Egiazarian, Petr Mokrov, and Evgeny Burnaev. Neural optimal transport with general cost functionals. In ICLR, 2024

    work page 2024

  2. [2]

    Engsig-Karup, George Em Karniadakis, and Cheol-Ho Jeong

    Nikolas Borrel-Jensen, Somdatta Goswami, Allan P. Engsig-Karup, George Em Karniadakis, and Cheol-Ho Jeong. Sound propagation in realistic interactive 3d scenes with parameterized sources using deep neural operators. Proceedings of the National Academy of Sciences, 2024

    work page 2024

  3. [3]

    Reusing the task-specific classifier as a discriminator: Discriminator-free adversarial domain adaptation

    Lin Chen, Huaian Chen, Zhixiang Wei, Xin Jin, Xiao Tan, Yi Jin, and Enhong Chen. Reusing the task-specific classifier as a discriminator: Discriminator-free adversarial domain adaptation. In CVPR, 2022

    work page 2022

  4. [4]

    Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems

    Tianping Chen and Hong Chen. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Transactions on Neural Networks, 1995

    work page 1995

  5. [5]

    Representation subspace distance for domain adaptation regression

    Xinyang Chen, Sinan Wang, Jianmin Wang, and Mingsheng Long. Representation subspace distance for domain adaptation regression. In ICML, 2021

    work page 2021

  6. [6]

    Optimal transport for domain adaptation

    Nicolas Courty, Rémi Flamary, Devis Tuia, and Alain Rakotomamonjy. Optimal transport for domain adaptation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2016

    work page 2016

  7. [7]

    Sinkhorn distances: Lightspeed computation of optimal transport

    Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In NeurIPS, 2013

    2013

  8. [8]

    Score-based generative neural networks for large-scale optimal transport

    Max Daniels, Tyler Maunu, and Paul Hand. Score-based generative neural networks for large-scale optimal transport. In NeurIPS, volume 34, pages 12955–12965, 2021

    work page 2021

Show all 40 references
  1. [9]

    Shaan Desai, Marios Mattheakis, Hayden Joy, Pavlos Protopapas, and Stephen J. Roberts. One- shot transfer learning of physics-informed neural networks. In ICML AI for Science Workshop, 2022

    2022

  2. [10]

    Neural monge map estimation and its applications

    Jiaojiao Fan, Shu Liu, Shaojun Ma, Hao-Min Zhou, and Yongxin Chen. Neural monge map estimation and its applications. Transactions on Machine Learning Research, 2023

    2023

  3. [11]

    Domain-adversarial training of neural networks

    Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario March, and Victor Lempitsky. Domain-adversarial training of neural networks. Journal of Machine Learning Research, 2016

    2016

  4. [12]

    Deep transfer operator learning for partial differential equations under conditional shift

    Somdatta Goswami, Katiana Kontolati, Michael D Shields, and George Em Karniadakis. Deep transfer operator learning for partial differential equations under conditional shift. Nature Machine Intelligence, 2022

    2022

  5. [13]

    Entropic neural optimal transport via diffusion processes

    Nikita Gushchin, Alexander Kolesov, Alexander Korotin, Dmitry P Vetrov, and Evgeny Burnaev. Entropic neural optimal transport via diffusion processes. In NeurIPS, volume 36, pages 75517–75544, 2023

    2023

  6. [14]

    Gnot: A general neural operator transformer for operator learning

    Zhongkai Hao, Zhengyi Wang, Hang Su, Chengyang Ying, Yinpeng Dong, Songming Liu, Ze Cheng, Jian Song, and Jun Zhu. Gnot: A general neural operator transformer for operator learning. In ICML, 2023. 10

    2023

  7. [15]

    Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries

    Junyan He, Seid Koric, Diab Abueidda, Ali Najafi, and Iwona Jasiuk. Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries. Computer Methods in Applied Mechanics and Engineering, 2024

    2024

  8. [16]

    Mionet: Learning multiple-input operators via tensor product

    Pengzhan Jin, Shuai Meng, and Lu Lu. Mionet: Learning multiple-input operators via tensor product. SIAM Journal on Scientific Computing, 2022

    2022

  9. [17]

    Optimal tensor transport

    Tanguy Kerdoncuff, Rémi Emonet, Michaël Perrot, and Marc Sebban. Optimal tensor transport. In AAAI, 2022

    2022

  10. [18]

    Neural optimal transport

    Alexander Korotin, Daniil Selikhanovych, and Evgeny Burnaev. Neural optimal transport. In ICLR, 2023

    2023

  11. [19]

    Transformer for partial differential equations’ operator learning

    Zijie Li, Kazem Meidani, and Amir Barati Farimani. Transformer for partial differential equations’ operator learning. Transactions on Machine Learning Research, 2023

    2023

  12. [20]

    Scalable transformer for pde surrogate modeling

    Zijie Li, Dule Shu, and Amir Barati Farimani. Scalable transformer for pde surrogate modeling. In NeurIPS, 2023

    2023

  13. [21]

    Fourier neural operator with learned deformations for pdes on general geometries

    Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural operator with learned deformations for pdes on general geometries. Journal of Machine Learning Research, 2023

    2023

  14. [22]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In ICLR, 2021

    2021

  15. [23]

    Learning transferable features with deep adaptation networks

    Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan. Learning transferable features with deep adaptation networks. In ICML, 2015

    2015

  16. [24]

    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, 2021

    2021

  17. [25]

    A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data

    Lu Lu, Xuhui Meng, Shengze Cai, Zhiping Mao, Somdatta Goswami, Zhongqiang Zhang, and George Em Karniadakis. A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. Computer Methods in Applied Mechanics and Engineering, 2022

    2022

  18. [26]

    Physics-informed neural network policy iteration: Algorithms, convergence, and verification

    Yiming Meng, Ruikun Zhou, Amartya Mukherjee, Maxwell Fitzsimmons, Christopher Song, and Jun Liu. Physics-informed neural network policy iteration: Algorithms, convergence, and verification. In ICML, 2024

    2024

  19. [27]

    Dare-gram: Unsupervised domain adaptation regres- sion by aligning inverse gram matrices

    Ismail Nejjar, Qin Wang, and Olga Fink. Dare-gram: Unsupervised domain adaptation regres- sion by aligning inverse gram matrices. In CVPR, 2023

    2023

  20. [28]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    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, 2019

    2019

  21. [29]

    Weatherbench: a benchmark data set for data-driven weather forecasting

    Stephan Rasp, Peter D Dueben, Sebastian Scher, Jonathan A Weyn, Soukayna Mouatadid, and Nils Thuerey. Weatherbench: a benchmark data set for data-driven weather forecasting. Journal of Advances in Modeling Earth Systems, 12(11):e2020MS002203, 2020

    2020

  22. [30]

    Large scale optimal transport and mapping estimation

    Vivien Seguy, Bharath Bhushan Damodaran, Remi Flamary, Nicolas Courty, Antoine Rolet, and Mathieu Blondel. Large scale optimal transport and mapping estimation. In ICLR, 2018

    2018

  23. [31]

    Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior

    Shashank Subramanian, Peter Harrington, Kurt Keutzer, Wahid Bhimji, Dmitriy Morozov, Michael W Mahoney, and Amir Gholami. Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior. In NeurIPS, 2023

    2023

  24. [32]

    Factorized fourier neural operators

    Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural operators. In ICLR, 2023. 11

    2023

  25. [33]

    ClimODE: Climate and weather forecasting with physics-informed neural ODEs

    Yogesh Verma, Markus Heinonen, and Vikas Garg. ClimODE: Climate and weather forecasting with physics-informed neural ODEs. In ICLR, 2024

    2024

  26. [34]

    A deep transfer operator learning method for temperature field reconstruction in a lithium-ion battery pack

    Yuchen Wang, Can Xiong, Changjiang Ju, Genke Yang, Yu-wang Chen, and Xiaotian Yu. A deep transfer operator learning method for temperature field reconstruction in a lithium-ion battery pack. IEEE Transactions on Industrial Informatics, 2024

    2024

  27. [35]

    Transolver: A fast transformer solver for PDEs on general geometries

    Haixu Wu, Huakun Luo, Haowen Wang, Jianmin Wang, and Mingsheng Long. Transolver: A fast transformer solver for PDEs on general geometries. In ICML, 2024

    2024

  28. [36]

    Interpretable weather forecasting for worldwide stations with a unified deep model

    Haixu Wu, Hang Zhou, Mingsheng Long, and Jianmin Wang. Interpretable weather forecasting for worldwide stations with a unified deep model. Nature Machine Intelligence, 2023

    2023

  29. [37]

    Transfer learning enhanced deeponet for long-time prediction of evolution equations

    Wuzhe Xu, Yulong Lu, and Li Wang. Transfer learning enhanced deeponet for long-time prediction of evolution equations. In AAAI, 2023

    2023

  30. [38]

    Cod: Learning conditional invariant representation for domain adaptation regression

    Hao-Ran Yang, Chuan-Xian Ren, and You-Wei Luo. Cod: Learning conditional invariant representation for domain adaptation regression. In ECCV, 2025

    2025

  31. [39]

    Skilful nowcasting of extreme precipitation with nowcastnet

    Yuchen Zhang, Mingsheng Long, Kaiyuan Chen, Lanxiang Xing, Ronghua Jin, Michael I Jordan, and Jianmin Wang. Skilful nowcasting of extreme precipitation with nowcastnet. Nature, 2023

    2023

  32. [40]

    Brady, Paul W

    Tingtao Zhou, Xuan Wan, Daniel Zhengyu Huang, Zongyi Li, Zhiwei Peng, Anima Anandkumar, John F. Brady, Paul W. Sternberg, and Chiara Daraio. Ai-aided geometric design of anti-infection catheters. Science Advances, 2024. 12 A Notations The notations appear in this paper are sum...

    2024

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.