Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning
Authors: Siyang Li
Yize Chen
Yan Guo
Ming Huang
Hui Xiong
Venue: ICLR 2026 Poster (2026)
Keywords: PDE Dynamics Forecasting
OOD Generalization
Invariant Learning
Primary area: learning on time series and dynamical systems
TL;DR: Introducing a physics-guided PDE invariance principle to achieve both superior in-distribution forecasting performance and zero-shot generalization capabilities on diverse OOD forecasting scenarios.

Abstract: Advanced deep learning-based approaches have been actively applied to forecast the spatiotemporal physical dynamics governed by partial differential equations (PDEs), which acts as a critical procedure in tackling many science and engineering problems. As real-world physical environments like PDE system parameters are always capricious, how to generalize across unseen out-of-distribution (OOD) forecasting scenarios using limited training data is of great importance. To bridge this barrier, existing methods focus on discovering domain-generalizable representations across various PDE dynamics trajectories. However, their zero-shot OOD generalization capability remains deficient, since extra test-time samples for domain-specific adaptation are still required. This is because the fundamental physical invariance in PDE dynamical systems are yet to be investigated or integrated. To this end, we first explicitly define a two-fold PDE invariance principle, which points out that ingredient operators and their composition relationships remain invariant across different domains and PDE system evolution. Next, to capture this two-fold PDE invariance, we propose a physics-guided invariant learning method termed iMOOE, featuring an Invariance-aligned Mixture Of Operator Expert architecture and a frequency-enriched invariant learning objective. Extensive experiments across simulated benchmarks and real-world applications validate iMOOE's superior in-distribution performance and zero-shot generalization capabilities on diverse OOD forecasting scenarios.

Published as a conference paper at ICLR 2026
TOWARDS GENERALIZABLE PDE DYNAMICS FORECASTING VIA PHYSICS-GUIDED
INVARIANT LEARNING
Siyang Li', Yize Chen", Yan Guo", Ming Huang", Hui Xiong 'Hong Kong University of Science and Technology (Guangzhou)
University of Alberta, 3 AAlibaba Cloud s11572@connect. hkust-gz. edu. en, mingqi an. hw@al -inc com, qingjian, -ywalibaba-ainnnnne inc. com xionghui gz edu.ccn
en, www..cceeeeuaeeeeee www.@ualberra..commm
ABSTRACT
Advanced deep learning- based approaches have been actively applied forecast the spatiotemporal physical dynamics governed by partial differential equations (PDEs), which acts as a critical procedure in tackling many science and engineer- ing problems. As real-world physical environments like PDE system parameters are always capricious, how to generalize across unseen out-tt-disttibuttin (OOD) forecasting scenarios using limited training data is of great importance. To bridge this barrier, existing methods focus on discovering domain genaaiiggeerrllzablle repre- sentations across various PDE dynamics trajectories. However, their zero-shot OOD generalization capability remains deficient, since extra test-time samples for domain-specifie adaptation physical invariance in PDE dynamical systems are yet to be investigated or inte- grated. To this end, we first explicitly define two-fold PDE invariance principle, which points out that ingredient operators and their composition relationships re- main invariant across different domains and PDE system evolution. Next, to cap- ture this two-fold PDE invariance, we propose a physics- sessidd-guiiedd invariant learn- ing method termed iMOOE, featuring an Invariance-alignan Expert architecture and frequency-enrichel invariant learning objective. Exten- sive experiments across simulated benchmarks and real-world applications vali- date iMO0E's 's superior in-distribution performance and zero-shot generalization capabilities on diverse OOD forecasting scenarios. The code is be publicly acces- sible at cattps://gitiiie www.thtobbcooo.ecceoooeeeeooe.ee wtt.ehww...cmmmmycccccccc.......
still required. This is because the fundamental
Mixture Of Operator
INTRODUCTION
Reasoning physical dynamics governed by partial differential equations (PDEs) is essential for wide range of science and engineering applications, such as meteorological prediction (Pathak et al.] 2022]. battery design (Wang et al.) (024a)) chemical synthesis (Gao & Günnemann 2024) and elec- tromagnetic simulation (Huang et al. 2022). As real-world PDE dynamical systems are always com- plex, ever-changing and even unknown, is difficult for traditional numerical methods explicitly discover the physical law, which requires intensive expert knowledge and computation resources. To this end. physics-infooe informed deep learning Yu & Wang 2024 Li et al. (024c) are applied identify unknown PDE dynamics and speed up calculation. For instance, neural operators (Kovachki et al.)
Li al., 2023c) are developed to discover the underlying PDE law based on observed trajec- tories and geometries. Score- based generative models (Li et al. 2024b [hhshheya et al.) 2024) are employed to reconstruct the full physical field from sparse measurements. Despite these success, the zero-shot out-of-distribution (OOD) generalization performance of PDE dynamics learning remains underexplored. It crucial to achieve accurate zero-shot PDE forecasting on unseen OOD scenarios without additional adaptation. It can obviate test-time retraining burden and accelerate various PDE system design and control problems (Hao et al. 2022).
2023
"Corresponding author.

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Published as a conference paper at ICLR 2026
To tackle OOD challenges in PDE dynamics learning, existing works focus learning domain- generalizable representations from multi donlt-ddmmann PDE dynamics. Such domain can be governed by variable physical parameters in PDE systems (Cho et al. 2024). This line of research can be cate- gorized into three classes. First, domain-aware meta-learning (Zintgraf et al. 2019) is leveraged to empower PDE forecasting models with fast adaptation ability test domains (Wang al. 2022b Kirchmeyer et al., [0222] Kassai Koupai et al. (2024). These methods divide the network parameter space into domain-invaan invariant and domain-specific parts, assuming they can represent shared and dis- tinct knowledge in parametric PDE systems. Second, parameter conditioning schemes (Takamoto al. 2023 Cho et al.] 2024, Gupta & Brandstetter, (2022) are developed and integrated into cur- rent neural PDE solvers, allowing them to generalize across varying parameters. Third, (Hao et al.) 2024a McCabe et al. 2024 Subramanian et al., 2023) demonstrate that pretraining on diverse PDE dynamics data can enhance the transferability to downstream forecasting tasks. However, the zero- shot OOD generalization capability of these methods is still lacking. They demand enough test-time samples and domain-speedt specific fine-tuning to achieve ideal performance. The core reason is that they do not explicitly illuminate the fundamental invariance principle across various PDE dynamics.
In this work, we look into the zero-shot generalizable PDE dynamics forecasting problem, with only limited variety of training trajectories available. This zero-shot setting excludes the access to test- time data for domain adaptation, which is essource-intensive and time-consuming. We are motivated by the invariant learning theory (Arjovsky et al. 2019 Liu et al. (202), which can provably achieve ideal OOD performance by exploiting the invariant correlations between inputs and targets across varying distributions. Although invariant learning has performed impressively on vision and graph OOD tasks (Liu al. 2022 Chen et al. [023a 2022), how to define and discover the basic physical invariance principle for OOD generalizable PDE dynamics forecasting remains unexplored. To this end, we propose address the zero-shot OOD forecasting problem by explicitly prescribing and estimating the PDE invariance from multiple training domains.
To bridge this gap, we first discover that for specific PDE system, there are two kinds of invariance independent of domain shifts: i) Individual physical processes dictated by a set of specialized oper- ators; ii) Composition relationships between these operators and exogenous conditions like physical parameters and forcing terms. For example, reaction-diffusion systems used in chemistry and ecol- ogy (Rao et al. 2023) consist of a diffusion process formed by Laplacian operator and a nonlinear reaction function, with diffusion and reaction coefficient controlling their rates respectively. The widely-used operator splitting method (Glowinski et al., 2017) for numerical PDE solving is built upon this discovery, which separates a complex PDE into several simpler operators and solves them by different numerical tools. Exploiting these two kinds of physics-guided invariant correlations can tackle the distribution shifts of PDE forecasting scenarios
zero-shot manner.
In this work, informed by the two-level invariance principle in PDE systems, we propose a physics- guided invariant learning method towards zero-shot generalizable PDE dynamics forecasting. Such PDE invariance learning can be realized by an invariance-alignen tive. Specifically, as PDE can be split into a set of compositional operators (Glowinski et al.) (017), we design mixture of operator experts architecture to capture these invariant operators and their composition relationships. It closely aligned with the proposed two-level PDE invariance. Then, we propose a celquueny-enriccer invariant learning objective to approximate the PDE invariance by equalizing the risk of various training domains. Our main contributions are summarized as follows:
network and risk equality objec-
We propose a physics-guided PDE invariance learning method termed iMOOE, which can achieve zero-shot PDE dynamics forecasting across diverse OOD scenarios.
A mixture of operator expert network and a frequency-augmmente risk equality objective are proposed to capture the two-fold PDE invariance.
Extensive experiments demonstrate superior zero-shot OOD generalization capability of iMOOE, as well as its delicate compatibility with diverse neural operators.
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2 OOD GENERALIZATION ON PDE FORECASTING
2.1 PROBLEM FORMULATION
In this work, we focus on forecasting the spatiotemporal dynamics of twwo-dimensional PDE systems which can be characterized in the following form:
0.u1 F (x, u1 U1 Ux ....u............ f.f), (l,x)) [0, x Ω
where x(t, R is system state variables defined within the time span T and spatial domain Q C IR2 indicates the PDE parameters that can reflect physical properties, such as the Reynold number in fluid dynamics. f denotes the forcing term from external input, such the heat source in the temperature field. F(-) represents the unknown PDE law that governs the underlying physical processes. 0x- the spatial derivatives which underpin the differential operators in F-)). Suppose we can collect system trajectories fu(t, 1= } of N₂ time steps from multiple environments Eall- The environment E Eaul can be distinguished by variable factors in PDE systems, which lead to diverse OOD scenarios. Akin to prior works on OOD dynamics forecasting (Liu et al. 2023). we consider distribution shifts on initial conditions u(0, x). physical parameters p. forcing terms f and temporal resolution N₁. We also assume periodic boundary conditions for PDE systems following et. 2021 Wang et al. (2024b Kassai Koupai al.) 2024). Under this multi-context setting, the goal of OOD generalizable PDE dynamics forecasting is to learn a neural simulator f()) on available CE.u> and f()) can perform well on all (unseen) domains without any test-time adaptation. This zero-shot OOD forecasting objective can be cast min-max risk optimization problem (Wang al. 2022a) below:
trajectories from limited training environments Di, (D" (D")) Ce
min max Re (f), S.t. R (f) = E. [e (f(1)) Y+),, f eEEan
where --= (u" (t,x) H-1 is a trajectory of past H steps observed from e, Y = (t,)) is the target sequence that should be predicted. E(-) is the loss function quantifying prediction errors. For brevity, we use to denote u(l, x) in the rest of content. We describe the practical value of such zero-shot OOD dynamics forecasting setting Appendix C.4
2.2 INVARIANT LEARNING FOR DYNAMICS FORECASTING
Directly solving the min- max optimization problem in Eq. is nontrivial. Following prior invariant leamning literature (Krueger et al.) 2021), we derive the optimal solution f by finding maximal invariant predictor which hinges on the invariant correlations between observed trajectories I and future targets Y. Let & and g denote PDE invariance extractor and output forecaster, then we can decompose f = g o Φ. In light of (Liu et al. 2021), the optimal d' (I) should satisfy two properties:
Sufficiency property: Y = "0" (I)) + e, where e is random noise. It requires (I) to possess sufficient predictive information that can forecast future dynamics Y.
Invariance property: Ey. (e (Ø (I,)), Y*) = -Ey [e(g" (T=( )), Y "), Ye, E Eaul
requires o² to identify the invariance principle in PDE dynamical system P(.). Such PDE invariance can give rise to equal risks across different forecasting environments.
Based on above two requisites for PDE invariance learning, the core challenge lies in how to identify the fundamental PDE invariance principle from multi-domain trajectories. The forecasting model f g" (Ø" (I)) built upon PDE invariance can realize the desirable zero-shot OOD performance.
2.3 INVARIANCE PRINCIPLE FOR PDE DYNAMICS
We now formally introduce the invariance principle for PDE dynamics forecasting. Due to the lack of such invariance formalism, domain-invarian representations learned by previous meta-learning- based (Kirchmeyer et al.) 2022 Kassai Koupai et al. 2024) parameter conditioning- based meth- ods (Takamoto et al. (2023) can not achieve ideal OOD outcomes. We define PDE invariance based on such finding: As the PDE law F(-) is composed of few operator items (Rudy al., 2017). the widely used operator splitting method can (Glowinski et al. 2017) decompose PDE into different operators and combine the solution of each part. It has exhibited great efficiency on wide range
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Past Trajetories
Future States
FRO-09000 stxpam2.7455
Ug
Up
Up
UT
1.10
Invariant
200
2.2
Operators (1)
Invariant Composition k(-)
2.0
nedd PS0
gi 1t
200
A((or:)
f
Ø
PS8 bendrre
Mera
Pronenier
0.0
0.0
6.04
(a) SCM for PDE-govereed dynamics.
(b) Case study on varying paramelers.
(C) ID-OOD correlations
Figure 1: (a) The SCM diagram for the formation process PDE dynamies. illustrates prescribed two-level PDE invariance and potential distribution shifts on exogenous inputs. (b) A case study by varying physical parameters DR dynamics to compare the zero-shot OOD performance of four methods. Without the guidance of formalized PDE invariance, previous methods can not achieve better OOD results on unseen environments.
(c) The ID-OOD correlations of two neural opera- tors. Based on the slope of two linear positive ID-OOD lines, FNO equipped with PDE invariance learning can capture more transferrable knowledge from limited training domains and achieve better OOD robustness. Refer to Appendix F6 for more details.
of PDE solving, including the complex nonlinear Navier-Stole Stokes equation (Glowinski et al.) 2017). Then, derive the two-fold PDE invariance principle which underpins PDE system evolution:
(i). Operator invariance: PDE dynamics are governed by the composition of a few spatial operators {o,(x, u, ..u,...)) )- These elementary operators representing distinct physics remain invariant across system evolution and different domains.
(ii). Compositionallity invariance: The composition method h to aggregate basic operators, physical parameters and forcing terms is fixed as F h(o...... T σK . f) for a specific PDE system.
f)dt + u, given the last observation u4- nutshell, invariant correlations between input and target involve: invariant operators {ax, ) invariant compositional relationships h among different items, as well as the fixed seppwwse numerical integration. We present a structural causal model (SCM) (Krueger cet al. 2021) in Fig. a to illustrate the formation process of PDE dynamics and its pertinent two- level invariant correlations. Regardless of various time steps and distribution shifts on [uo, p. f). the fundamental set of operators and their composition relationships
In addition, the future state Ūc+1 can be calculated û,+1 - h((a,,
remain invariant. We formally justify the proposed two-level invariance principle can hold across many specialized PDE systems from three perspectives in Appendix reaction coefficient of DR dynamics in Fig. b), and the ID-OOD correlation lines (an effective met- ric to assess OOD robustness Yuan et al. 2023) in Fig. e).. Both of them can further demonstrate the effectiveness of the proposed physics-guide PDE invariance learning for improving zero-shot OOD capability. Refer to Appendix C for more related works on PDE dynamics forecasting and invariant learning.
B Besides,
provide case study by varying diffusion and
PHYSICS-GUIDEE INVARIANT LEARNING FRAMEWORK
To develop the physics-guided invariant learning for zero-shot OOD forecasting, the key challenge resides in how to cultivate an effective invariant forecaster that can exploit two-level PDE invariance principle defined in Sec. 2.3 To achieve this, we first design mixture of operator experts network which can respect the invariant correlations between past observations and future trajectories. In vision OOD tasks, the mixture-of-experts (MoE) architecture has shown great generalization, since MoE can closely align with the invariant correlations between image attributes and labels (Li al.) 2023a). But how to enable MoE to capture PDE invariance for zero-shot OOD forecasting remains an open issue. Next, propose a frequency-enriched invariant learning objective to estimate PDE invariance from multiple training domains. It can tackle the high-frequency learning pitfall in ex- isting neural operators (Khodakarami et al. 2025) Up now, derive the invariant Mixture
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of Operator Experts (iMOOE), a physics guiiid invariant learning method towards zero-shot OOD generalizable PDE dynamics forecasting as depicted in Fig. 2
3.1 ARCHITECTURE ALIGNMENT: MIXTURE OF OPERATOR EXPERTS
To align with the operator and compositionality invariance presented in Sec. 2.3 we develop the MOOE architecture which consists of two parts: i) A group of specialized neural operator experts to represent the unique and unknown physies. ii) A fusion network to aggregate these expert output with exogenous input like system parameters. This design shares the similar spirit with the effective operator splitting solver (Glowinski et al. 2017), which separates a complex PDE into a set of sim- pler operators and calculates each part by suitable numerical methods. Taking the ceactionddiffusion equation as an example (Krishnapriyan et al. 2021), , we can solve the second-order diffusion com- ponent by finite difference and calculate the reaction function by forward pass. Note that the typical operator splitting algorithm for PDE solving has a serial structure, which treats the solution of the former operator as the initial condition of the latter operator. But such serial operator solving will lead to slow computation for neural PDE learning. In this regard, the developed MOOE network stacks operator experts in parallel instead of linking them in series, as depicted in Fig. a).
Operator Experts. Similar to operator splitting, each individual operator expert in MOOE should be specialized in approximating a distinet physical process. In addition, we observe that each oper- ator component in PDEs is formed by state variable or its certain orders of partial derivatives. For instance, advection term this regard, we design a binary mask vector m = {0, 1] for each operator expert oi, making it can adaptively select the useful spatial derivatives to benefit operator learning. Here, the number of pre-computed derivatives 0x= u. We leverage existing neural operators (Kovachki et al. 2023) as the backbone operator experts, which excel at approximating PDE laws:
consists of 0 whereas diffusion term V stems from 0xx u. In
a, = =N: (x, up-W m, @ [0x Oc. 1F).
(3)
A notable advantage is that expert NO; can be compatible with a broad variety of neural operators without any modification on their structures. We verify this compatibility in Section 4.3 Apart from spatial coordinates x and past sequences of length W,, we also incorporate pre-calculated derivatives as prior input, which can render operator learning easier (Li et al. 2024a). To encourage operator experts to represent inhomogeneous physical processes, we feed them with different sets of pre- calculated derivatives by designing a mask diversity loss:
K2 EE exp (xp(ellm (-11m:-m,11)
Lmask
min (m.)
We illustrate this masking-bases input derivative selection design in Fig. b). and analyze its effect in Appendix F.1.2.
Fusion Network. The key role of the fusion network aggregate the output operator experts and condition it on physical parameters in variable p and f. Regarding the parameter conditioning, we directly concatenate the expert output with PDE parameters and utilize a multi-layer perceptron (MLP) to encode it. As for the aggregator, we should consider two different cases after some empir- ical trials: i) For PDE systems with strong non-linearity, such as the convection term u V(V x u) in turbulence flow (Dresdner et al., 2023), we employ an extra network to learn this complex compo- sition. ii) For PDE systems without these intractable non-linest terms, such as the additive operator relationship in reaction- reaction-diffusion. representing the invariant composition relationship h can be expressed as follows:
simply add up expert output. In brief, the fusion network
NutionNet (MLP) (o,, p. f),,.... MLP (aK, f)).
(5)
In Appendix F.1.3
we verify that choosing suitable type of fusion network can better align with the compositionality invariance and achieve better OOD performance.
3.2 FREQUENCY-EENRCHEDD INVARIANT LEARNING OBJECTIVE
With the MOOE network which can align with two-fold PDE invariance, the next step is to design an effective invariant learning objective that can satisfy two requisites presented in Sec. 2.2 However,
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Input 1 Waik
Operator Ex
[p'.r
Forecast: Biil
R
pend
Operator Expert
o(()
Ra
Derivative Selection mm,
Off Shalf Neural Operators
Input 1 Wo-2
Operator Expert Ci;
Fusion Network
C Come
ANO
DeesOlte
Perecast: uf
CC Crng
Transform en based
Operator Expert
Neural Field-bease
P
(#) IMOOE frumework.
(b) Individual operator experts.
Figure 2: (a) Overview of iMOEE method, which can capture the physics-guided PDE invariance by the mixture of operator experts architecture and frequency-enricher multi-context ((Er, = 2 here) training. (b) The structure of single operator expert, which well fit in diverse neural operators.
we find that the intrinsic spectral bias issue of neural operators will hinder PDE invariance learning due to the neglect of high-frequency information. To mitigate it, we propose a -egqueeecy-augmeneed invariant learning loss which can help to capture the complete PDE dynamies.
representations
Maximal Prediction Loss. Following invariant learning literature (Liu et al. 2021), we can fulfill the sufficiency property by maximizing the mutual information between the two-level PDE invari- ance h {a: 1 and future forecasts Y In light of (Tsai et al. 2021). this information maximiza- tion objective can realized by the maximal prediction loss dynamics forecasting:
[ M((or)EE: (.)d.
Ppred
Σ eEEer
Ryreds and R Rppeed
Exp
Σ
|Etrl
where we utilize the autoregressive training manner and ú is the predicted state at the last time step. We utilize the Euler forward method to implement the numerical integration on time marching.
Risk Equality Loss. The invariance property demands prediction error across various environments be equal. As proved in invariant learning literature (Krueger al. (021), can meet this risk equality objective by minimizing the variance of risks over different training environments:
Line Var ((Rpredd cee.))
where Roved is provided in Eq. 6 We borrow the useful linear scheduling scheme to impose this risk equality loss (Krueger et al. 2021). which reserves an initial empirical risk minimization stage (i.e. pretraining stage merely by Lpred) to learn the rich predictive representations. Refer to Appendix F.1.5) for the effect of this linear invariant loss scheduling. Note that our design differs from (Krueger al. 2021) in environment division. In addition taking physical parameters in and as environment labels, we also partition training domains by different autoregressive steps, since there exist covariate shifts in p[I, Y Specifically, during the autoregressive prediction, the distribution of past sequences p(I") can change with the time marching, but the correlations between Ie and Y keep invariant at each time step. We find this step-wise division is instrumental for fluid dynamics forecasting such as Navier-Stokes and Burgers systems, as shown in Appendix F.1.4
Frequency Enrichment Loss. Both Lpred and Line are inadequate to capture the complete PDE invariance, since neural operators prioritize learning the dominant low-frequency features in state (a.k.a. the spectral bias issue) (Lippe al. 2023). Ignoring the necessary high-frequency modes entails spectral information loss for invariant operator learning, which impedes to satisfy the sufficiency property given in Sec. [2.2] Besides, figh-frequenyy learning errors can propagate the whole spectral domain during the autoregressive prediction process, rendering hard to generalize across OOD scenarios with different frequency distributions. To this end, we propose to augment high-frequenee representations when learning PDE invariance by designing regularization item:
N
Lireq
Rtror and Rr = Ep. eEErr
ΣD
Σ
IEL (u.) (E) ((.) (5)
(8)
|Etrl
where FF is the fast Fourier transform and & is the wavenumber vector for each spatial frequency. Apparently, the weight | can pay more attention the high-frequency modes each forecasting

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step. We validate such frequency enrichment loss induce better PDE forecasting generalization in Appendix. F.I.I Prior works on OOD vision recognition (Chen et al. 2023a Zhang et al. 2022)
also proved that diverse and rich features can lead to better OOD capability
3.3
OVERALL FRAMEWORK
The total PDE invariance learning objective for iMOOE is presented below:
Ltotal = XpredCpredd + Xine + freg + Amask
Equipped with this hybrid training loss and invariance allined architecture developed in Sec. 3.1 we can effectively learn the proposed PDE invariance to achieve zero-shot OOD forecasting. Existing neural operators always train with prediction loss Epred. without any effort to learn the fundamental PDE invariance principle. This could be the key reason for their failures on OOD dynamics fore- casting. We demonstrate in Section 4.3 that when equipped with the explicit physics-informed PDE invariance learning method iMOOE, current neural operators Moreover, in Appendix [.7 we further investigate how the properties multi-environment training data can affect the zero-shot OOD capability of iMOOE. As simulating PDE trajectories or measur- ing real-world PDE dynamics is expensive, such analysis can provide a guideline on how to collect training data under limited data budget.
realize better OOD performance.
EXPERIMENTS
4.1
EXPERIMENTAL SETUP
Datasets. We adopt five PDE dynamical systems in different fields for the spatiotemporal physical dynamics forecasting task:
-eassionn (DR) (Takamoto et al.] 2022). Navier-Stokes (NS) et (021), Burgers (BG) (Hao et al.) 2024b), Shallow- Mater (Takamoto et al. (022) and Heat- Conduction (HC) (Hao et al., 2024b). We construct a wide range of OOD scenarios by varying the physical parameters of initial conditions ug. PDE coefficients p. forcing terms f or temporal resolu- tions N,. and OOD parameters for PDE simulation are randomly drawn from two non-overlapped uniform distributions, while previous parametric PDE learning works like (Kassai Koupai et al., 2024 Takamoto et al. 2023) just select several separate parameters. The spatial resolution of each state frame is fixed to 64 x 64. See Appendix D for detailed description data generation.
Evaluation Criteria. We leverage two metrics in PDEBench (Takamoto et al. 2022) to comprehen- sively evaluate the forecasting performance: i) normalized Mean Squared Error (nMSH) in raw data space: nMSE- || uH-N, 112 || 1l.. ii) fourier Root Mean Squared Error (IRMSE) in
frequency domain: fRMSE = v E Ean
((( - F(uH.N / (Smax - Emin + 1). They can reflect the forecasting accuracy of PDE system states from both data and physics views. Note that for all experiments, we present both in-distribution (ID) and out-of-distributtin (OOD) results in zero-shot setting (i.e. without any access to test-time samples for adaptation).
Ear
Implementation Details. We fix the number of operator experts K and loss weights Apred = 1, 1,rer = 001, Amask 0.001. Similar to prior invariant learning work (Krueger et al. 2021), linearly schedule Xino with an upper threshold of 0.001. The popular Fourier Ncural Operator (FNO) (Li et al. (021) with 4 layers and 64 width is employed
the backbone of each operator expert. We pre-calculate first- and second-order spatial derivatives for adaptive selection by masking. Past
HI W 10 steps observations are used predict future trajectories, following the same setting in previous PDE forecasing works (Li et al., [202]] Kassai Koupai et al.] 2024). iMOOE is trained on a NVIDIA A100 GPU with total 500 epochs, 0.001 initial learning rate by Adam optimizer.
4.2 ZERO-SHOP OOD PERFORMANCE
Baselines. We select six latest PDE forecasting methods with highlighted OOD generalization capa- bility: i) CoDA (Kichheeyer et al. 2022) and GEPS (Kassai Koupai et al. 2024): two context- meta-learning-bart based models. iii) CNO (Raonic et al. (023): a robust convolutional neural operator. iv) DPOT (Hao et al.
CAPE (Takamoto et al. 2023): parameter conditioning method.
2024a):
transformer- saadeermer-baeed based operator with denoising pretraining. v) VCNeF (Hagnberger et al. 2024
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Table I: Zero-shot ID/OOD generalization results compared to existing generalizable PDE dynamics forecasting methods. The listed five PDE dynamical systems are employed to synthesize a diversity of OOD forecasting scenarios. Best results are in bold and second-best results are underlined. "n.a." indicates the excess of computational resource limit.
Metrics
Models
DR
NS
BG
SW
HC
ID
00D
ID
0OD
ID
OOD
ID
0OD
ID
OOD
CoDA
3.40e-1
6.05e-1
4.31e-1
9.14e-1
8.72e-1
9.22c-1
D.a.
D.a.
1.16e40
2.37e40
CAPE
3.90e=
7.16c-2
9.09e-2
3.56e-1
5.00e-3
3.04e-2
2.71e.6
6.18e:5
5.70e-2
3.65e+0
CNO
3.36++0
2.56c+0
6.03e-1
6,90c-1
3.30e-3
1.87e22
2.10e-5
3.82e-5
1.01e-1
2.45c+0
iMSE
DPOT
2.62c-2
5.67e-2
3,44c-1
5.08e-1
2.18c-2
8.41e-2
6.69c-5
4.85c-4
3.68c-2
2.12c40
VONelf
8.70e-3
7.84e-2
1.40e-1
3.81e11
1.03e-2
4.68e-2
4.59e.5
6.12e-4
1.37e40
1.42e40
GEPS
8.71e:3
7.94e-2
2.07e-1
4.13e-1
2.24c-2
7.56e-2
1.22e-4
2.76e-4
9,43e-1
1.35e+0
IMOOE
5.15e-3
4.23e-2
6.49c-2
3.12e-1
1.20e-3
1.08e-2
3.34e-7
3.02e-5
3.92c-2
1.22c+0
CoDA
7.88e-3
9.93e-3
3.81e-2
7.31e-2
2.12e-2
2.50e-2
n.a.
n.a.
1.25e-2
7.09e-3
CAPE
1.1c-3
2.21e-3
1.97-2
5.77e-2
2.13c-3
5.70e-3
1.22c-4
5,50e-4
2.05e-3
8.83e-3
CNO
3.06e-2
2.43e-2
4.67e-2
7.79e-2
2.60e-3
5.82e-3
3.35e-4
4,79c-4
2,84--3
7,54c-3
IRMSE
DPOT
3.00e-3
2.80e-3
4.59e-2
7.30e-2
5.32e-3
1.04e-2
6.61e-4
1.95-3
3.08e-3
7.83e-3
VCNeF
1.68c-3
2.77e-3
2.66e-2
6.11e-2
3,20c-3
7.28c-3
5.46o-4
1.3e-3
1.26e-2
7.13e-3
GEPS
1.996-3
3.28c-3
3,85e-2
6.85e-2
5.14c-3
9.38c-3
9.08c-4
1.38e-3
1.04e-2
6.16c-3
iMOOE
9.16e-4
1.78e-3
1.38e-2
5,36e-2
1.10e-
3.83e-3
4.59e.5
3.65e-4
1.31e-
5.95e-3
Table 2: Zero-shot time extrapolation results on two PDE systems.
DR
NS
Models
nMSE
IRMSE
nMSE
IRMSE
In-time
Out-time
In-time
Out-time
In-time
Out-time
In-time
Out-time
CAPE
3.30c-3
3.88c-1
1.11c-3
8.58c-3
1.92c-1
5.46e-1
4.36c-2
7.04c-2
DPOT
4.51e-3
4,58c+0
1.31e-3
1.75e-2
1.94e-1
6.22e-1
4.62e-2
7.88e-2
VCNeF
2.41e-3
5.46e-1
9.81e-4
1.26e-2
3.00e-1
9.67e-1
4.90e-2
8.66e-2
GEPS
2.52e-3
6.96e-1
1.02e-3
1.39e-2
2.70e-1
6.57e-1
4.78e-2
7.77e-2
iMOOE
9.47c-4
1.99c-1
4.93e-
6.26e-3
166e-1
4.57e-1
3.89--2
6.79c-2
conditional neural field-based method. Note that meta-learning-basee methods commonly require few-shot adaptation to perform OOD forecasting. In Appendix F.4] we describe how to adapt them to zero-shot setting and further compare zero-shot iMOOE with few-shot CoDA, GEPS. Implemen- tation details of these baseline models are provided in Appendix [99
Results. We report ID/OOD generalization outcomes on various unseen scenarios in Table [ It is obvious that iMOOE can achieve the state-of-art (SOTA) results this simulated benchmark, with an average increase of 40.21% on nMSE and 30.78% on TRMSE. Such considerable promotion re- flects that explicitly learning the proposed physics-guided PDE invariance can boost zero-shot OOD performance on PDE dynamics forecasting. Moreover, we present the OOD results on extrapolated temporal resolutions in Table 2] Following previous time extrapolation setting (Kassai Koupai al., 2024], we train on [0, N₂] and test on [0, 2Nr). We find that iMOOE can achieve SOTA results with an average growth of 32.51 % on nMSE and 15.30% on fRMSE. indicates that learning the un- derlying PDE invariance across time steps can improve the OOD performance unseen temporal distribution shift scenarios. To measure iMOOE's zero-shot OOD capacity more clearly, we present an empirical upper bound for its OOD performance in Appendix F55
4.3 UNIVERSALITY STUDY
In Table 3 we manifest iMOOE's flexibility on integrating diverse operator learning models into op- erator experts (--) in plug-and-play fashion. We involve four classic categories of neural operators VCNeF (Hagnberger et al. 2024) and Transformer-based OFormer (Li et al. (023b). We validate their vanilla capability and iMOOE-upgraeed performance on DR dynamics under 8 OOD environments, and present OOD nMSE results of each environment in Table 3 Existing neural operators have not been comprehen- sively validated under this zero-shot OOD setting. When augmented by either MOOE or iMOOE, these neural operators can consistently achieve lower mean and variance values nMSE over var- ious OOD contexts. Such promotion underscores both the PDE invariance-aligned architecture and frequency-enriched objective can improve zero-shot OOD capability of existing neural operators.
including FNO (Li et al. 202T), DeepONet (Lu al. 2021). neural
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Published as a conference paper at ICLR 2026
Table Operator compatibility study on DR data with various OOD contexts. "EnvI" "Env8" in- dicates eight different settings for diffusion and reaction coefficients. "+MOOE" denotes employing vanilla neural operators as the backbone of operator experts. "+iMOOE" denotes further imposing the frequency-enrichel invariance training MOOE.
Operators
Variants
Env
Eav2
Eav3
Eov4
Env5
Env6
Env?
Eav8
Mean
Std
Naive
6.78e-2
8.80e
6.70e-2
6.00e-2
3.14e-2
1.11e-1
1.62e-1
4.68e.2
7.94e:2
3. 3.88e-2
+MOOE
3,40e-2
6.00e-2
3.88e-2
3.28e-2
1.88e-2
8.14
1.22c-1
2.49c-2
5.16e-2
3.26c-2
+MOOE
3.19c-2
5.20c-2
3.12c-2
3.05e-2
1.4c-2
6.02c-2
9.93e-2
1.87c-2
4.23e-2
2.60e-2
Naive
5,94e-1
9.73e-1
6.38e-1
5.18e-1
3.99; 1
7.27e-1
5,58e 1
5.09e
6.15e-1
1.63e1
NeepONet
+MOOE
6.03e-1
9.74e-1
3.70e-1
7.18e-1
5.56e-1
4.82e-1
6.10e-1
1.69e-1
+MMOE
5,45e-1
8,75c-1
5,76e-1
4,93c-1
3,47c-1
6.31e-1
4.82c-1
4,45c-1
5,49c-1
1.47e11
Naive
5.77e-2
9,74e-2
6.11e-2
5.35e-2
2.48e-2
1.26e-1
1.68e-1
3.85e-2
7.84e-2
4,54e-2
VCNeF
+MOOE
3.51e:2
6.91e:2
3.37e:2
3.31e:2
2.62e-2
8.86e
3.31e:2
5.73e-2
3.72e-2
*MOOE
3,29e-2
6,59e-2
3.01e-2
3,30e-2
2.60e-2
8.31e-2
1,37e-1
3,36e-2
5.52e-2
3.62e-2
Naive
4,95c-2
6.74e-2
4.75e-2
4.74c-2
5.27c-2
6.77c-2
7.47c-2
5,31e-2
5.75e-2
1.01o22
OFoemer
MOOE
4.47e-2
4.38e-2
4.16e-2
4.98e-2
5.31e-2
4.78c-2
6.60e-2
5.03e-2
4.96e-2
7.12c-3
+MOOE
4.06e
4.15e:2
4.17e:2
4.80e
5.29e
3.47e:2
3.65e
5.09e.2
4.34e-2
6.18e-3
Target
IMOOE
DyAd
CAPE
Table 4: Zero-shot OOD results on SST dynamics.
Dayst
nMSE
IRMSE
Models
Meun
Std
Meun
Std
GEPS
8.24e-1
3.08e-1
5.28e-2
6.02e-3
Days5
DPOT
5.56e-1
2.43c-1
3.65e:2
5.19c-3
VCNeF
6.69c-1
2,67e-1
4.61e-2
6.39e-3
DyAd
5.87c-1
2.44e-1
3.79c-2
5.15e-3
CAPE
6.51e-1
2.81e-1
3.84e-2
5.59e-3
HMSE-0.0259
nM56-0.0691 nM5E-0.1341
IMOOE
5.12e-1
2.36e-1
3.44e-2
5.03e3
Figure 3: Test SST sample showcase.
4.4 APPLICATION TO REAL-WORLD PDE DYNAMICS
Apart from the simulated benchmark above, we also leverage two real-world PDE-governed Ocean- Atmosphere dynamics datasets, including Sea Surface Temperature (SST) (Huang al. 2021 and stereo Sea Surface Elevation (Guimarães et al. 2020) to validate iMOOE's generalization capability. These two datasets represent upper-ocean thermodynamies and free-surface ocean wave dynamics respectively, and contain sensory measurement noise. In SST forecasting setting, a specific region on Pacific Ocean selected and divided into 60 60 grid. We predict SST state of future 6 days using past 4 days observations. SST data between year 1982-2019 and 2020-2021 is utilized for training and testing. Note that each independent SST trajectory can be deemed as an instance from unique environment, since daily SST variations are affected by many meteorological conditions like solar radiation and wind speed. Input parameters are also unknown so we feed one-valued vector to the fusion network. In SSE forecasting setting, choose the wave dynamics recorded at La Jument lighthouse with total 4500 frames on 241 x 221 grid. The training and testing sets are formed by the first 4000 frames and remaining 500 frames. We input past 4 steps SSE state predict future steps. We take a typical ocean dynamics forecasting baseline called Dy Ad (Wang et al.) (202b)) and present OOD comparison results Table 45 We find that iMOOE can attain the lowest mean and variance on two metrics across various OOD samples, which reflects iMOOE's capability to capture the underlying physics law in real-world ocean dynamics. Test samples in Fig. 3 3 exhibit iMOOE can capture local variations of SST and SSE with higher fidelity and accuracy. Beyond 2D PDE- governed dynamies, we further demonstrate iMOOE's zero-shot OOD capability can be extended to other types of dynamical systems in Appendix F2
Table 6: Effect of varying numbers of operator experts.
Number of expert K
DR
BG
Inference time
mMSR
IRMSE
nMSH
IRMSE
5.26c-2
1.94c-3
1.38e-2
4.35e-3
0.08s
2
4.63e-2
1.81e-3
1.14c-2
3,95e-3
0.1
3
4.17e-2
1.74e-3
1.07e-2
3.83e-3
0.15s
4
5.00e-2
1.84e-3
1.14c-2
3.88e-3
0.18s

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Published as a conference paper at ICLR 2026
Target
ByAd
Table 5: Zero-shot OOD results on SSE dynamics.
Models
nMSI:
IRMSE
Mean
Mean
Std
GEPS
3.41e-2
4.96e-3
4.11e:3
1.55e-4
DPOT
2.46e:2
3.28e:3
3.54e:3
1.49c-4
Last
VCNeF
3.41e-2
4,96c-3
4.11c-3
1.55e-4
DyAd
3.17e-2
4.61e-3
3,96e-3
1.50c4
CAPE
3.34e-2
4.85e-3
4.07e:3
1.53e-4
nMSE-00.0098 nM5E-0.0290 nMSE-0.0313
IMOOE
1.52e-2
2.34e-3
2.78e-3
1.42e44
Figure 4: Test SSE sample showease.
4.5 SENSITIVITY ANALYSIS
In Table we investigate the influence of the number of operator experts K on iMOOE's zero-shot OOD performance. To ensure fair comparison, we only escalate K from I to 4 and keep other setups like the width of FNO and training batch size unchanged. We find that the best-performing group is K 3 while the worst setting is K = 1. This reveals that small K (i.e. only expert) is not sufficient to capture the operator invariance, while large K (i.e. 4 experts) could be redundant given that actual PDE systems contains only few number of compositional invariant operators (Rudy et al. 2017). Besides, the increasing number of neural operators can exacerbate the computational overhead. Refer to Appendix for more detailed explanations the effect of expert number K and its distinction from the mixture-of-expert (MoF) architecture large foundation models (LFMs).
Refer Appendix F3 for more sensitivity analysis loss weights in Eq. 9
CONCLUSION
In this work, we propose the iMOOE learning framework to address the zero-shot OOD general- ization issue in the scope of PDE-governed spatiotemporal physical dynamics forecasting. We first introduce the two-level mhysics-guided invariance principle for PDE dynamical systems. Then, we develop the mixture of operator experts architecture plus the gaguunccy-uumenedd invariant learning objective to capture such PDE invariance from limited training environments. Various experiments demonstrate the excellent zero-shot OOD forecasting capability of iMOOE. However, the proposed PDE invariance learning is validated on a limited diversity of dynamical systems. In future work, we plan to extend iMOOE's zero-shot OOD capability other types of PDE dynamics, such as PDE systems on irregular grids, or more real-world applications like earth system forecasting.
ETHICS STATEMENT
Our work is only aimed at generalizable PDE dynamics forecasting for human good, so there is no involvement of human subjects conflict of interests far the authors are aware of.
ACKNOWLEDGEMENTS
This work was supported in part by the National Key R&D Program of China (Grant No. No.2023YYFF022001) in part by the National Natural Science Foundation of China (Grant No.92370204), in part by the guangdong Basic and Applied Basic Research Foundation (Grant No. No.2023B15151220057, part by the Key-Area Special Project of Guangdong Provincial Ordinary Univessites(20442IZXX10777
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A
LLM USAGE
Large Language Models (LLMs) were used to aid in the writing and polishing of the manuscript. Specifically, we used an LLM to assist in refining the language, improving readability, and ensuring clarity in various sections of the paper. The model helped with tasks such sentence rephrasing, grammar checking, and enhancing the overall flow of the text. It is important to note that the LLM was not involved in the ideation, research methodology, experimental design. All research con- cepts, ideas, and analyses were developed and conducted by the authors. The contributions of the LLM were solely focused improving the linguistic quality of the paper, with involvement in the scientific content or data analysis. We have ensured that the LLM-generated text adheres to ethical guidelines and does not contribute to plagiarism or scientific misconduct.
B RIGOROUS JUSTIFICATION FOR TWO-LEVEL PDE INVARIANCE
Firstly, in the field of PDE learning, the commonly utilized assumption is that many specialized PDE systems demonstrate compositional structures and contains group of distinct physical pro- cesses, as claimed in Section 4.1 of (McCabe et al., 2024). For example, Navier-Stokes systems are composed of convection, pressure and viscosity, wave systems are composed of Hamiltonian flow and dissipation. This PDE compositionality assumption can be formulated (Xu et al. 2025): 0,u(x) = F (o) (u, .... (u, x), ..., dK (u, x)). F(-) is a nonlinear function, which represents the compositional relationship among a group of basic operators {a; x) x)} Such compositional operator structure guides us to design the two-level PDE invariance principle. The proposed iMOOE
leverages a fusion network to approximate F()) and a set of neural operator experts to approximate (u,
Secondly, as observed in classical PDE discovery literature such as SINDy (Brunton et al. 2016) and PDE-FIND (Rudy et al. 2017). PDE systems of interests can be naturally represented in a few spatial operators. These PDE discovery methods resort to SVD-based linear combination of few terms in the fixed operator library. This indeed motivates us to design the operator invariance principle, and also justifies the compositional structure of operator learning.
Thirdly, we justify the composition of neural operators (i.e. iMOOF's architecture design) can ap- proximate arbitrary continuous operators according to the universal approximation theorem proved in (Kovachki et al. (023). As illustrated in Section 9.3 and Figure 16 in (Kovachki et al. (023)) ) neu- ral operators that can approximate continuous operators should possess three modules: i) Mapping input to the first hidden representation by a local operator. ii) Mapping each hidden representation to next by a non-local integral kernel operator. iii) Mapping the last hidden representation to output by local operator. In practical implementation, i) and iii) can be realized by local neural network like MLP, while ii) should be realized by a global neural network like transformer or FNO layers. The proposed composition of neural operators realizes i) and ii) by concatenating the last hidden representations of multiple neural operators, and realizes iii) by the fusion network. Hence, iMOOE architecture design can also satisfy the universal approximation theorem of operators.
C RELATED WORK
C.I SPATIOTEMPORAL PDE DYNAMICS FORECASTING
Deep bearning-based dynamics forecasting centers around developing diverse neural operators to decipher the unknown time-dependent PDE systems. This line of research has been extensively leveraged to reason a wide scope of real-world spatiotemporal dynamics. like atmospheric circula- tion (Pathak et al., 2022), ocean wave (Cui et al. 2025), turbulent fluid (Xing et al. 2024), and power system transient (Cui et al. 2023). Their innovations come from either new operator architectures or more robust autogressive training methods, which are aimed at addressing a range open issues in PDE forecasting, including solving parametric PDEs (Takamoto et al. 2023 Cho et al. 2024), full-field reconstruction from sparse observations (Shysheya et al. 2024 Li et al. 2024b). irregular
geometries (Li et al.) 2023c Wu et al. 2024a) long temporal process stability (Lippe al. 2023
Rühling Cachay et al. 2023). However, most of these works do not highlight the zero-shot OOD generalizable forecasting, which is significantly crucial problem for two reasons: i) The unseen
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OOD scenarios always occur in real-world PDE dynamics prediction, owing to the ubiquitous distribution shifts of forecasting contexts, encompassing system parameters, external forcing func- tions, initial conditions and sampling conditions. ii) As procuring abundant dynamics trajectories to learn leabe representations is expensive, many PDE dynamics forecasting methods cultivated in low-data regime. In this sense, how to generalize across diverse OOD environments with limited training data is of great importance, Although several studies have explored the poten-
tial of meta-learning (Kirchmeyer al. 2022; Kassai Koupai al. 2024) parameter conditioning Takamoto et al. | G & Brandstetter (2022) methods in OOD forecasting, their zero-shot generalization capability remains lacking since they can not expose the truly fundamental invariance in PDE dynamical systems. Another drawback is that they need to carry out ad-hoc modifications to current neural operator architectures. To remedy them, we elucidate the physical invariance principle of PDE dynamics from two perspectives, and then develop a tixture-of-Expee (Mof))-based archi- tecture which can delicately integrate existing operator learning methods to capture PDE invariance in plug-and-play manner.
Note that the MoE-based architecture (Dai et al. 2024) has been extensively employed Large Language Models (LLM) increase the representation capacity and knowledge density without sacrificing the inference speed. Few neural PDE solvers based on spatial domain decomposition (Hao et al. 2023; Chalapathi et al. (224) also borrow this parallel structure improve the compu- tational efficiency for large-scale PDEs. It shares the same spirit with the finite element method, as each expert is assigned to calculate on a sub-domain and coordinating these experts can behave well on complex geometries. In contrast, our proposed mixture of operator expert architecture is aimed capturing the domain-invariant operators for zero-shot generalizable forecasting. Another differ- ence from LLM on MoE usage is that experts in LLM are sparsely activated according to the routed token, whereas MoE in neural PDE is always dense as each expert should account for sub-operators or sub-domains.
C.2 INVARIANT LEARNING FOR OOD GENERALIZATION
Invariant learning Arjovsky et al., 2019: Liu et al. 202T) is an effective paradigm to boost OOD generalization performance. It aims to discover invariant representations that can possess sufficient information to predict targets and elicit equal risks across various (unseen) environments. There exist two open issues in invariant learning: i) how to prescribe the domain-specific invariance prin- ciple for different learning problems; ii) how to design the effective OOD objectives to estimate the defined invariance on limited training contexts. Existing works strive to address these challenges from different aspects, such feature learning (Chen al. 2023a), multi-objective optimization (Chen et al., (0236)) architecture alignment (Li et al. 2023a). information bottleneck (Ahuja et al. 2021] and gradient consistency (Rame et al. (022). These research outcomes have been success- fully applied to vision recognition (Li et al.] (222a)). molecule prediction (Chen et 2022 Li et al.)
2022). pedestrian motion forecasting (Liu et al. (202)) and time series analysis (Liu et al. (024))
However, the efficacy of invariant learning for PDE dynamics forecasting is still under-explored. To bridge this gap in this work, we propose to unleash its power by fostering an iMOOE architecture and optimizing it by rreuency-enriihhr risk equality loss, both of which can help to capture the complete PDE dynamics invariance.
C.3 PHYSICAL INVARIANCE LEARNING
Incorporating physical prior knowledge into deep learning is a valid way to improve the general- ization capacity, data efficiency as well as the physical consistency of produced predictions (Yu & Wang 2024). light of this, a line of related research focus on imposing domain-specific physics knowledge which remains invariant in PDE dynamical systems, for the sake of better accuracy and
OOD robustness of dynamics reasoning. These physical invariance can involve symmetries (Wang al. 2021), conservation laws (Huang et al. 2024), exact physics models (Holt et al. (024) or basis function dictionaries (Mouli et al.) 2024). In this work, we propose a two-level invariance princi- ple for PDE dynamics inspired by the formation process of PDE laws and useful operator splitting method. Such PDE invariance can be deemed as a kind of prior physical knowledge, which need to be digged out by physics-informed invariant learning.
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C.4 PRACTICAL VALUE OF HERO-SHOT OOD DYNAMICS FORECASTING
In the scope of both parametric PDE simulation and real-world PDFE-governed physical dynamics forecasting, the zero-shot OOD generalization is a ubiquitous and urgent issue. i) In many industrial manufacturing fields which require high-intensity PDE calculation, such as electromagnetic simu- lation (Huang et al., (022) and airfoil design (Wu et al. 2024b). PDE parameters are ever-changing due to the varying material properties and ambient factors. is also hard acquire valuable test- time trajectories for each new physical environment Thus the zero-shot OOD simulation is highly demanded. ii) In many spatiotemporal physical dynamies forecasting fields such as weather and climate prediction (Bodnar et al.) 2025), there always exist unforeseen dynamics patterns in me- terological variables due to the chaotic nature of systems and unpredictable human activities. is impossible to collect abundant training contexts which can cover all the unforeseen test scenarios. It also computational expensive to fine-tune the weather foundation model (Bodnar et al., 2025) for the hourly or daily inference. Thus the zero-shot OOD forecasting is greatly significant.
D MULTI-ENVIROMMENT DATASET DETAILS
Unless otherwise stated, the experiments conducted in this work follow the same data setting during the training and test stage: i) Training data: 16 environments with 64 trajectories per environment. ii) Test data: 16 environments with 8 trajectories per environment. All OOD forecasting experiments are executed in zero-shot settings, without any test-time samples for model fine-tuning or adaptation. Below, we clarify the multi-context state trajectory generation method on five woo-dimensional PDE dynamical systems. We assume the boundary conditions (BCs) are fixed (e.g. periodic BC) for each PDE system, so that BCs are not regarded as environment variables.
Diffusion-Reactt (Tkamooto et al. 2022). The underlying DR equation is presented as:
0,u D, u + Dr + (u- u² - k 1-2) u))
(10)
01e D,O + Dy + (u (u-v).
(11)
v, v denote the concentrations of activator and inhibitor respectively. The spatiotemporal domain is
(x,t) [0. x [0, 20). At the initial state, two objects are randomly localized into six 0.2 0.2 squares. Two diffusion coefficients D.. D. and one reaction coefficient k are assigned to construct different physical contexts. The physical parameters of training trajectories are drawn from DN [1c-3, 2c-3]. De E [5e-3, 1e-2], k E [5e-3, 1e-2], while the OOD test parameters are fetched from D. E [2e-3, 3e-3), D. E [le-2, 12. k E [le-2, 1.5e-2]. We utilize past 10 steps sequence to forecast future 11 steps states, i.e. H = W- 10 and N, = 21.
Navier-Stokes (Li al. 2021). The vorticity-type incompressible NS equation is presented as:
Pnw 11 V + vAw + 0.1 (sin(wr(x + y)) + cos(wr(< + v))), V 0. (12)
w. denote the velocity field and fluid vorticity. The spatiotemporal domain (x, E [o, 1J3 x [0, 50). The initial vorticity are produced from normal Gaussian random field. The viscosity co- efficient v and the frequency coefficient w in the forcing term can be employed to generate diverse physical environments. We simulate training sequences using E [le-5, 1e-3] and OOD test se- quences using / E [5c-6, 8c-6] U [1.2e-3, 2e-3] with a fixed w = 2. We utilize previous 10 steps trajectories to forecast future 21 steps vorticity, i.e. H = W 10 and N, 31.
Burgers (Hao et al. 2024b). The coupled BG equation is presented as:
d = u- Vu + rdu.
(13)
u denotes the fluid velocity field. The spatiotemporal domain is (x,t) E [0, 64 x [0, 1]. Identical
to (Hao et al.) 2024b), we also adopt the same sine and cosine functions over the spatial domain to generate initial conditions. The viscosity coefficient is employed to produce diverse forecasting scenarios. The training sequences sequences are simulated from // E [2.5e-3, 3, 4e-3] U [6e-2, 1. The past 10 steps series are utilized to forecast future 11 steps velocity, i.e. H W 10 and N, 21.
simulated from // E [5c-3, 5e-2], whereas the OOD test
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Shallow-Water
TTakmmoto et al., 2022). The hyperbolic SW equation is presented as:
0yh 8,hu Gyh 0,
(14)
0, + 02
1² + 2
9
= h,h
(15)
Ochv + 0g 12 + g h²
hr
(16)
u, v denote the velocities along the horizontal and vertical axis. h denotes the water depth and b is a spatially varying bathymetry. hu, he can be perceived as the directional momentum components. Gr indicates the acceleration of gravity. The spatiotemporal domain is (x, () E [0, 5]2 x [o, 1]. Akin to (Takamoto et al., 2022), the initial conditions are shaped as 2D radial dam breaks. We take their initial radius as physical parameters to construct data contexts. The training series are fetched from radius within [0.3, 0.63], and the OOD test series are obtained from radius within [0.63, 0.7.. The prior 10 steps series utilized to forecast the water depth of future 11 steps, i.e. H W 10 and N, = 21.
(Hao et al. 2024b). The HC equation with varying heat source is presented as:
0,u V (a(x) + A sin(m1 sin(m2 sin sinmm)tt).
(17)
denotes the temperature field over the spatiotemporal domain (x, L) E [0, 1]2 x [0, 5]. Similar to (Hao et al. 2024b), the coefficient function a(x) is stipulated as exponential Gaussian random
field. The external forcing terms are altered to generated various physical contexts. We specifically vary three frequency coefficients m,, M2, my of heat sources and keep the amplitude A = 200. The training temperature fields are produced by m1, my [1, 2], m2 E [5, 10], and the OOD test fields stem from m1, my E [2. 3]. m2 E [10, 15]. We utilize past 10 steps fields to forecast the temperature of future 11 steps, i.e. H W 10 and N₂ 21.
E
MORE ANALYSIS ON MIXTURE OF OPERATOR EXPERT ARCHITECTURE
E. DISPARATE MoE USAGE IMOOE AND LFMS
The key difference lies in during the forward pass, LFMs (Dai et al. 2024 Liu et al. 2025 Shi et al., 2025) need to selectively activate a sparse number of FFN experts, while IMOOE stands for dense version of MoE which should aggregate the output of all neural operator experts by the designed fusion network. Commonly, LFMs require a huge number of experts to express the fine- grained and specialized knowledge in pretraining data corpus, and their performance on downstream tasks can benefit from the large capacity of specialized experts. However, as for PDE invariance learning, the invariant knowledge is prescribed as the composition of a few number of invariant operators, since real-world PDE dynamical systems often consist of a small set of physical processes (Rudy al., (017). Accordingly, iMOOE can capture the underlying PDE law by only few number of operator experts. Besides, LFMs usually enable each FFN expert to represent distinct knowledge by the load balance loss (Dai et al., 2024). While iMOOE leverages the proposed mask diversity loss to adaptively select different sets of spatial derivatives for expert input, which can explicitly enforce individual operator experts to express distinct physical processes.
E.2 DETAILED EXPLANATIONS ON THE EFFECT OF THE NUMBER OF OPER ATOR EXPERTS
The mixture of operator expert architecture specifically designed to closely align with the pro- posed two-level PDE invariance principle. In Table 6 we can observe it is not strict that iMO0E's zero-shot OOD capability can constantly promote with the increase of the expert number K for two reasons: i) Overfitting risk. A large K will increase iMOOE's model complexity. When K is overly large but the operator invariance is not that complex, such as K = 4 for DR dynamics, iMOOE is likely to overfit to the limited training domains (i.e. 16 training environments with to 1024 DR trajec- tories). It can diminish the accuracy and robustness of captured PDE invariance. ii) Representation redundancy. Real-world PDE systems are usually composed by a few number of physical processes, such as the DR system only contains a Laplacian operator and a reaction function. A overly large K could render the representations of these FNO experts redundant each other. For example,
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when we input second derivatives [uzz: Urz Myy. Tyy: May. Vay to four FNO experts to learn
DR dynamics, their actual learned masks are m, = [0, 1, 1, 0, 1, m2 = [0. 1, 1, 1, 0, 1], ma = [1, O, 0, 0, 1, 11, m, [0, 0, 1, 0, 0, 0j. We can observe that the first and second expert behaves very similarly to each other, and the fourth expert unnecessary since its behavior can be covered by other three experts. Thus K can perform better than K = 4 as shown in Table 6
F ADDITIONAL RESULTS
F.1 ABLATION STUDY
F.1.1 EFFECT OF FREQUENCY-ENRICHER LOsS IN EQ. 8
We investigate the benefits of the proposed frequency enrichment loss Lyreg for PDE invariance learning. We utilize the simulated DR data and real-word SST, SSE data to validate the improved forecasting generalizability induced by additional regularization on figh-freeuency representations. Diffusion and reaction coefficients can dictate the distribution of frequency patterns in DR evolu- tions. SSE contains high-frequency short waves which could be caused by the nonlinear surface features like wave breaking fronts, sharp crests and bound harmonics. SST contains seh-frrquency modes due to the ocean advection and vertical processes such as upwelling. Apart from nMESE and IRMSE metries, we also present the forecasting errors within different frequency bands in Table "Low", "Mid", "High" denote non-overlapped ranges of wavenumber ę: Elow E [0, 4J. Emid E [5, 12]. Sligh E [13, {maz]. When equipped with freeurryy augmentation, the ID/OOD nMSE can drop by 24.38% and 25.00%, and ID/OOD fRMSE can decrease by 10.20% and 12.32% on DR data. also improves nMSE by 9.22% and 8.98%, fRMSE by 7.03% and 7.33% for SST, SSE data. Notably, improving high-frequency feature learning can also enhance the OOD accuracy on both low-frequency and mid-frequency patterns. Such ID/OOD promotion verifies the necessity of the proposed requency-enricche objective, which can mitigate the spectral bias of neural operators and help to capture the complete PDE invariance from the spectral domain.
Table 7: Ablation results of frequency enrichment loss on DR data.
ID
0OD
Methods
FRMSF
IRMSF
RMSE
nMSE
Low
Mid
Hlgh
Total
Low
Mid
High
Total
w/o Liree
6-81e-3
3.39e-3
1.17e-3
3.31e-4
1.02-3
5.64e-2 4.23e-2
9.29e-3
1.25e-3
4.42e-4
2.03e-3
w/ Creg
5.15e-3
2.96e-3
1.09e-3
3.03e-4
9.16e-4
7.92e-3
1.17e-3
4.20c-4
1.78e-3
Table 8: Ablation results of frequency enrichment loss on real-word ocean dynamics data.
SST
SSE
Methods
IRMSE
IRMSE
aMSE
nMSE
Low
Mid
High
Total
Low
Mid
High
Total
w/o Liven
5.64e-1
1.29e-1
3.55e-2
9.24e.3
3.70e-2
1.67e-2
3.03e-3
2.98e-3
3.01e-3
3.00e-3
w/ Lirra
5.12e-1
1.24e-1
3.06e-2
8.47e-3
3.44e-2
1.52e-2
2.10e-3
2.74e-3
2,94e-3
2.78e-3
F.2 EFFECT OF PRE-CALCULLATED DERIVATIVE SELECTION
We investigate the effect of input spatial derivative selection designed in Section 3.1 incorporates certain orders of pre-calculated spatial derivatives into each operator expert input and specific mask diversity loss which can encourage experts to represent distinct operators. We report the influence of this derivative selection design in Table 9 We can find that especially for the real- world SST changing dynamics which are complex and hard to capture, the prior derivative imput can make it easier and more accurate to discover SST's physical law. (Li et al. 2024a) consistently val- idated that introducing additional spatial derivatives can improve neural PDE learning. We conduct further analysis on this design follows:
Such design
i) Effect of mask diversity loss mmask- We set Amask = 0 and feedforward all pre-computed deriva- tives each expert. The OOD nMSE and FRMSE results on DR data are 4.78e 2 and 1.89e 3,
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leading to 13.0% and 6.18% degradation versus standard iMOOE. We find masks learned by two experts are similar to each other, which hinders them from representing distinct invariant operators.
ïi) Effect of derivative types. As BG equation contains first-order and second-order derivatives, take both of them as prior input for vanilla iMOOE, and each mask learns to adaptively select the needed derivatives for its coupled operator expert. But when we just input first-order derivatives, OOD nMSE and fRMSE increase to 1.16e and 3.85e - 3, with 7.41 % and 0.52% degradation. This verifies that prior second-order derivatives can improve learning efficiency for BG systems.
iii) Actual learned mask vectors m. When learning on DR data, we feed second-order derivatives (Unaary Manes Myyy Pyy. Mayy Pryl to two experts in iMOOE, and their actual learned mask is m
[0, O, 1,,1,,1,1 0J. m2 = 1, 0, 0, 1]. As there are many operator splitting methods for DR equation (e.g. dividing into diffusion and reaction terms is just one of them), iMOOE can learn suitable splitting way via learning operator invariance from limited data.
Table 9: Ablation results of input spatial derivative selection.
DR
BG
SST
Methods
nMSE
IRMSE
nMSE
IRMSE
nMSE
IRMSE
w/o derivative selection
4.95e-2
1.95e-3
1.13e-2
3.94e-3
6.07e-1
3.82e-2
w/ derivative selection
4.23e-2
1.78e-3
1.08e-2
3.83e-3
5.12e-1
3,44e-2
Degradation, 17.02%
9.55%
4.63%
2.87%
18.55%
11.05%
F.1.3 EFFECT OF THE CHOICE OF FUSION NETWORK
We verify that properly choosing the type of expert fusion methods (presented in Section 3.1 is crucial to learn the accurate PDE invariance. We can determine the type of fusion network in light of prior physical knowledge on PDE systems. To focus on this network structure study, we abandon additional multi-environment invariance training. We take DR and NS systems for comparison and provide OOD results in Table TO We can find that for linear PDE systems such as DR, which holds simple additive relationship between the diffusion operator and reaction function, simply summing up the outputs of operator experts is a better fit. But for strongly non-line non-linear PDE systems like NS, which include complex operator multiplication, we should impose extra fusion network and let learn how to integrate expert outputs to capture the non-linear PDE law.
Table 10: Ablation results of the choice of two types of fusion methods.
DR
NS
Expert Composition Methods
nMSE
FRMSE
nMSE
FRMSE
Linear fusion by simple addition
5.80e-2
2.02e-3
4.82e-1 3.76e-1
6.32e-2 5.54e-2
Non-linear fusion by extra network
6.46e-2
3.28e-3
F.1.4 EFFECT OF Two ENVIRONMENT PARTITION METHODS
As mentioned in Section 3.2. dividing the training environments based on autoregressive time steps can further boost the outcomes of PDE invariance learning. We adopt two fluid dynamics datasets to verify the benefit of this step-wise partition method in addition common parameter- brased division. During the fluid evolution, state variations on two consecutive time steps are quite distinct, but the physics transition law between these two steps remain invariant. Therefore, we can regard each autoregressive step as a unique context. In Table [1] we present the effect of two environment partition methods. We can see that combining two partition methods together can realize the best ID/OOD performance, since it can enhance the diversity of training environments and improve the robustness of learned PDE invariance representations. Besides, parameter-base based partition performs moderately better than step-wise partition, as different physical parameters can lead more distinct PDE trajectories, such as the Reynold number in NS is a decisive factor to distinguish laminar or turbulent flow.
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Table 11: Effect of environment partition methods on fluid forecasting.
NS
BG
Partition methods
ID
OOD
ID
OOD
nMSH
FRMSH
nMSH
FRMSH
nMSH
FRMSH
nMSH
FRMSH
Only purameters
7.11e-2
1.50e-2
3.41e-1
5.49c-2
1,48c-3
1.17e-3
1.1le-2
3.94e-3
Only time steps
7.39c-2
1.52e-2
3.58e-1
5.52e-2
1.57e-3
1.19e-3
1.18e-2
4.01e-3
Parameters+time steps
6.49e-2
1.38e-2
3.12e-1
5.36e-2
1.20e-3
1.10e-3
1.08e-2
3.83e-3
F.1.5 EFFECT OF LINEAR Loss SCHEDULING
According to previous invariant learning implementation (Krueger al. 2021). the linear scheduling scheme is an effective and canonical way to impose the risk equality loss Live neural networks. To probe its effect on PDE invariance learning, we compare the performance of the MOOE model with fixed Line or linearly added Live in Table [2]] Concretely, "fixed" means Line keeps at 0.001 during the whole training procedure. "Linearly scheduled" indicates Line zero during the ini- tial 175 epochs, then linearly increases to 0.001 during the intermediate 150 epochs, and finally stays at 0.001 during the last 175 epochs. The main distinction between these two schemes lies in whether executing traditional empirical risk minimization (ERM) training by the maximal predic- tion loss Cpred during the initial pretraining stage of 175 epochs. Prior invariant learning works (Chen et al. 2023a Zhang al., 2022) claim that native ERM pretraining can help to gain rich data representations at the beginning. Invariant learning can be deemed as a certain way to filter out the Nomain-generalle
representations, We verify its effect DR dynamics as shown in Table [2 We can find that compared to fixing Line from scratch, linearly imposing Line on MOOE can lead better ID/OOD forecasting accuracy and lower error variance across test environments. It reflects that linear scheduling scheme can be better way to conduct the PDE invariance learning objective when training on diverse physical environments.
Table 12: Ablation results of linear invariant loss scheduling on DR data.
ID
0OD
Methods
nMSE
IRMSE
nMSE
IRMSE
Mcan
Mcan
Mcan
Std
Mean
Std
MOOE+fixed Linv
5.61e-3
4.93e-3 4.84e-3
1.04c-3
3.91e-4
5.80e-2
7.59e-2
2.02e-3
4.76e-4
MOOF++inearly scheduled C....
5.06e-3
9,23e-4
3.86e-4
4.93e-2
5.77e-2
1.87e-3
4.06e-4
F.2 APPLICABILITY TO DIVERSE DYNAMICAL SYSTEM FORECASTING
In this section, we demonstrate the proposed physics-infommn informed invariant learning method iMOOE can be casily extended wide variety of dynamics forecasting scenarios, apart from the 2D PDE systems on regular grids. In the following, we validate iMOOE's zero-shot OOD forecasting perfor- mance on neural ODE systems, 3D fluid dynamics and real-world time series. We can adapt iMOOE to these diverse dynamics by replacing the expert backbone with task-specific architectures.
F.2.1 APPLICATION TO ODE-GOVERNED DYNAMICS FORECASTING
We verify iMO0E's zero-shot OOD capability on neural ODE systems, as prior meta- meta-learning-bars based methods such as CoDA (Kirchmeyer et al. 2022) and GEPS (Kassai Koupai et al., (024) are also extended to Oov-goeeee dynamics forecasting. We conduct simulation on a typical ODE system called damped and driven pendulum equation, as shown in Appendix B.1 of (Kassai Koupai et al.) 2024). We utilize past 10-step states to forecast the pendulum motion angle of future steps. The time horizon of collected trajectories is [0, 25] and N₂ 51. We construct 16 ID training domains and 8 OOD test domains by randomly drawing four ODE parameters from the ID/OOD ranges given in Table [3] Identical to CoDA and GEPS, 4-layer MLP network with 64 hidden dimension is taken as the backbone for operator experts iMOOE. The pre-calculated spatial derivatives and mask diversity loss are discarded since they are unnecessary for ODE simulation. We report OOI)
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forecasting performance of zero-shot iMOOE and few-shot CoDA, GEPS in Table 14 iMOOE can achieve 10.4% and 12.66% decrease on nMSE and fRMSE versus GEPS. This may stem from their difference on discovering physical invariance. Specifically, iMOOE explicitly prescribes the two-level invariance principle and directly captures it via the proposed physics-informet invariant learning. While hypernttorrk-baser meta-learning methods like CoDA, GEPS estimate such in- variance by implicitly operating in the network parameter space without any physical guidance.
Table 13: ID/OOD parameter ranges of pendulum
Table 14:
OOD forecasting results on ODE- governed pendulum dynamics.
system for environment generation.
Parameters
ID Range
OOD Range
Models
nMSE
IRMSE
Damping coefficient a
[0.1,0.2]
[0.2,0.31
CoDA 5,31e+0
5,96e-2
Natural frequency 4h
[0.5,1.0]
[1.0,1.5]
GEPS
2.50e+0
5,45e-2
Forcing frequency
[0.3,0.6]
[0.6,0.9]
iMOOE
2.24e+0
4.76e-2
Forcing amplitude F 10.1,0.21
[0.2,0.31
F.2.2 APPLICATION TO 3D FLUID DYNAMICS FORECASTING
Apart from the typical 2D PDE dynamics, we also demonstrate iMO0E's zero-shot OOD perfor- mance on 3D PDE systems. We employ the 3D compressible Navier-Stokes equation in PDEBench Takamoto et al. 2022) and construct ID and OOD scenarios using the same method in Appendix D Specifically, for shear and bulk viscosity coefficients, we still randomly draw their values from the ID) parameter range [1e - 5, le -e--3 3] and OOD) range [5e- 6, Se 6] U [1.2e 2e 3, 2e - 33. The Mach number is kept as 1.0. The number of ID training and OOD test domains as well as their data volume
are also identical to setups in Appendix [D] The size of 3D spatial domain is 32 x 32 x 32, and the past 10-step velocity field sequences are provided to forecast the future 1 steppp states. FNO3d is utilized as the backbone of operator experts for iMOOE3d. We present the 3D OOD dynamics forecasting results in [55 We observe that iMOOE can attain 17.36% and 39.26% decrease on nMSE and IRMSE compared to previous 3D neural operators. Such results further validate the effectiveness of proposed physics-guided PDE invariance learning to more complex 3D PDE dynamics.
F.2.3 APPLICATION TO REAL- WORLD TIME SERIES PREDICTION
We further validate IMOOE's OOD capability on real-world time series prediction. Such time series dynamics are hard to be directly parsed by ODE or PDE laws. We leverage the Electricity Trans- former Temperature (ETT) data (Zhou et al. 2021) and follow the data split setting in (Zhou et al.) 2021). The changing dynamics of Oil Temperature (OT) is hard to decipher. OT dynamics are asso- ciated with exogenous covariates like electricity load. The task is to predict future 96-step OT values given lookback 512-step OT and six auxiliary power load sequences. To implement iMOOE this task, we borrow the Moirai-MoF. (Liu et al. (025) as backbone to approximate the invariant knowl- edge in ETT dynamics. The proposed fregueney-augmented invariant learning objective is utilized to fine-tune Moirai-MoE. As temporal distribution shifts are ubiquitous in time series domain (Kim) al. 202T), we simply deem each segmented ETT trajectory an independent environment. Con- sequently, iMOOE can attain 25. 25.29% and 13.59% growth on nMSE and fRMSE compared to native Moirai-MoE as reported in Table 16
Table 15: ID/OOD forecasting results on 3D NS dynamics.
Table 16: OOD forecasting results ETT time series dynamics.
aMSE
IRMSE
Models
Models
nMSE
fRMSE
OOD
ID
OOD
UNet3d
1.67++0
1.92++0
3.09e-1
5.16e-1
Informer
3.17c-1
1.07c-2
FNO3d
3.83e-1
1.88c40
4.99c-2
1.84c-1
Moirai-MoE
5.26e-2
4.93e-3
VCNeF3d
1.69c-1
6.97c-1
5.88c-2
1.35c-1
iMOOE
3.93e-2
4.26e-3
iM00E3d
1.19c-1
5.76e-1
2.11e-2
8.20e-2
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F.2.4 APPLICATION TO PDE DYNAMICS ON IRREGULAR SPATIAL DOMAIN
We leverage the Airfoil forecasting benchmark on non-uniform grid proposed in OFormer (Li et al., 2023b), where the underlying grid is divided by highly irregular triangular meshes. This Airfoil dataset contains 1000 training and 100 testing sequences with different inflow speed (Mach number) and angles of attack. The objective is to jointly predict future 22 steps velocity, density and pressure using past 4 steps states. One of the key advantages of iMOOE is the elegant compatibility across various neural operators. Hence, iMO0E can integrate neural operators that are able to handle irreg- ular spatial domains or unstructured meshes well, such as OFormer (Li et al. 2023b), Geo-FNO (Li et al. 2023c) and VCNeF (Hagnberger et al. 2024). We can easily enable iMOOE on irregular geometries by two tiny adaptations: i) The prior spatial derivatives on irregular grid are calculated by the finite element method. ii) The frequency enrichment loss relying on fast fourier transform on uniform grid is discarded. In Table [7] we present that when equipped with the proposed iMOOF method, three neural operators can exhibit better OOD forecasting accuracy on flows around Airfoil.
We randomly showcase a OOD flow velocity forecast in Fig. 13 16
Table 17: OOD forecasting results on irregular spatial domains.
Operators
Variants
nMSE
RMSE
Naive +iMOOE
5.45e-2 5.08e-2
2.22e-1 1.98e-1
VCNeF
Naive
5.15e-2
2.16e-1
Geo-FNO
+iMOOE
4.35e-2
1.88e-1
Naive
4.95e-2
2.04e-1
OFormer
+iMOOE
4.14e-2
1.79c-1
F.3 MORE RESULTS ON SENSITIVITY ANALYSIS
By hyperparameter tuming, we empirically find the fixed setting Apred = 1.0, Ann= = 0.001, Ayreg 0.1, Amask = 0.001 can perform well on both simulated and real-world physical dynamics data. As the effect of L.nask has been discussed in Appendix F.1.2. we investigate iMOOE's sensitivity to different Airee and free values. OOD results in Table 18 reflect that we should assign moderate values to Xinu and Xreg for satisfactory outcomes.
i) Sensitivity to invariance loss weight Aina. For a smaller Xine 0.0001, it diminishes the power of Line to capture the physical invariance and renders iMOOE overfit to training environments. For larger ALn 0.01, the degradation stems from the intrinsic conflict between Live and Lpred according to Section 3 in REx (Krueger et al. 202I). REx claims that overly minimizing the variance of errors across training domains can increase the error of the best-performing domain.
ii) Sensitivity to frequency loss weight Afreg- For a smaller Xfrey = 0.01, the generalization errors caused by high-frequency pitfalls can not be mitigated. For larger À freg 1.0, the high- shhhfrruy modes are over-optimized but the dominant low-frequency modes are not learned well.
Table 18: Influence of loss weights in Eq.
nMSS
IRMSE
Loss
Asne
Axreg
ID
OOD
ID
OOD
0.01
0.1
5.58e-3
5.15e-2
9.26e-4
1.94c-3
Lyne
0.0001
0.1
5.40e-3
5.24e-2
9.12
1.98e-3
0.001
1.0
5.75e-3
5.04c-2
9,30c-4
1.95e-3
C freq
0.001
0.01
5.42c-3
4.89c-2
9. 9.18
1.88c-3
Ours
0.001
0.1
5.10e-3
4.23e-2
9.16e-4
1.78e-3
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F.4 FURTHER COMPARISON WITH META-LEARNING- BASED METHODS
We first clarify how to adapt meta-learning-bee based baselines including CoDA (Kirchmeyer et al. 2022) and GEPS (Kssaai Koupai et al. 2024) to zero-shot OOD forecasting, Both CoDA and GEPS separate their network parameter space into domain-invariant and domain-specific parts. Domain- specific parameters need to be independently trained within each unique environment and require few-shot adaptation. When applied to zero-shot OOD testing, domain-invarian parameters keep freezing, while domain-specific parameters including 0F in CoDA and c in GEPS are initialized by averaged parameters over diverse training domains (e.g. Ctr Te.. Σ Eeel c9). This test-time ini- tialization method for domain-specific parameters is directly borrowed from GEPS (Kasaä Koupai et al. 2024), as stated in its last paragraph of Section 4.1.
Furthermore, implement CoDA and GEPS through their vanilla few-shot adaptation manner and our zero-shot inference setting. For few-shot setting, as claimed in Section 5.2 of both CoDA
(Kirchmeyer et al. 2022) and GEPS (Kassai Koupai et al., al.]|2022) (2024), we draw only one PDE trajectory
from each unseen test environment to finetune the domain-specific network parameters. In Table 19 we present OOD forecasting outcomes of zero-shot iMOOE and zero/few-shot CoDA and GEPS. It apparent that iMOOB can outperform CoDA and GEPS with few-shot test-time adaptation, due to their difference on discovering PDE invariance. Specifically, iMOOE explicitly prescribes the two-
level PDE invariance principle and effectively approximates it via the proposed physics-informed mixture of operator expert architecture and invariant learning objective. While meta- veeta-learningbbeee based CoDA, GEPS assume PDE invariance lies in domain-invarian. network parameters. They implicitly learn donaanngenerrlizzalee representations in the parameter space without any physical guidance.
Table 19: Further comparisons with meta-learning- -baed methods.
BG
NS
Methods
nMSE
IRMSE
nMSE
IRMSE
CoDA(zero-shot)
9,22e-1
2.50e-2
9.14e-1
7.31e-2
CoDA(few-shot)
6.89c-1
1.84c-2
6.31c-1
6.45c-2
GEPS(zero-shot)
7.56c-2
9.38c-3
4.13c-1
6.85e-2
GEPS(few-shot)
5.37c-2
6.58e-3
3.32c-1
5.47c-2
iM00E(zero-shol)
1.08e-2
3.83e-3
3.12e-1
5.36e-2
F.5 EMPIRICAL UPPER BOUND OF ZERO-SHOT OOD PERFORMANCE
To more intuitively gauge iMOOE's zero-shot OOD forecasting capability, we propose verify the empirical upper bound for iMOOE's OOD performance. Such bound can measure iMOOE's achievable OOD performance on unseen domains with distribution shifts (Gagnon-Audet et al. 2023). Akin to the test operation in (Gagnon-Audet et al. 2023), we randomly select four OOD test domains in DR data and train the standard FNO under each specific environment with differ- ent volumes of training trajectories. In Table 20 we compare the OOD performance of zero-shot iMOOE with three levels of empirical upper bounds forged by FNO. Apparently, iMOOE can con- sistently surpass the 16-shot FNO and rival the 64-shot FNO, while underperforming the 256-shot FNO. further demonstrates the proposed PDE invariance learning can improve the zero-shot OOD capability of neural operators unseen scenarios.
Table 20: Empirical upper bound of zero-shot OOD capacity on DR data.
E
Env2
Env3
Env4
Models
nMSE
IRMSE
nMSE
FRMSE
aMSE
IRMSE
nMSE
fRMSE
FNO(256-shot)
3.40e-3
8.90e-4
3.65e-3
6.56e-4
1.38e-3
4.09e-4
4.59e-3
7.69e-4
FNO(64-shoo)
6.38c-2
3.71e-3
6.20e-2
1.95e-3
1.8e-2
1.23e-3
4.34e-2
2.36c-3
FNO( 16-shot)
2.10c-1
6.71e-3
9.41e-2
3.30e-3
4.06c-2
2.08c-3
1.27c-1
3.95e-3
FNO-IMOOE//eno-shot)))
1.27c-2
1.65e-3
5.12c-2
1.82c-3
6.10c-2
1.42c-3
4.30c-2
2.09c-3
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F.6 MORE RESULTS ON ID-OOD CORRELATIONS IN FIG. [c)
In the scope of domain generalization, ID-OOD correlation (Miller et al. 2021 Yuan et al. 2023) useful metric to reflect the effective OOD robustness of deep learning model. If the relation- ships between ID and OOD test errors are sharply positive (i.e. the slope of ID-OOD fitted line positively large), we can claim that the developed neural network indeed captures the domain- generalizable representations from training data and its OOD robustness is satisfactory. In practice, the ID-OOD correlation line can be obtained by testing the developed model under various train- ing hyper-parameters, such as changing the quantity of training data, total epochs, initial learning rates, etc. For example, single blue scatter in Fig. 5 represents the FNO-IMOOE model with unique training configuration. The same interpretations for the orange scatter of FNO. As the slope of FNO-iMOOE's ID-OOD line significantly sharper than that of FNO, we can state that when FNO is augmented by the proposed PDE invariance learning framework, it is able capture the fundamental invariance in PDE dynamics and achieve better OOD forecasting performance.
0.0050
FNO slope=0.7843
FNO-IMOOE slope=1.1345
0.0045
0.0040
OOD test FRMSE
0.0035
model FNO-IMOO
0.0030
FNO
0.0025
0.0020
0.0015
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
test TRMSE
Figure 5: Supplementary IRMSE results for ID-OOD correlations.
F.7 ANALYSIS ON TRAINING DATA PROPERTIES
In practice, either measuring real-world dynamics trajectories by multi-source sensors generating simulated PDE data by numerical solvers is prohibitively expensive. To this end, it is of great signif- icance to investigate the impact of training data properties zero-shot OOD forecasting capability. This can guide us to construct more informative multi-context sequences and further improve OOD performance from the data perspective. We conduct this study by answering is the effect of training data quantity? ii) When the budget of collecting training data is limited, in terms of data diversity (i.e. the number of training environments |E 1) and data quantity within each environment, which factor is more important? DR data is utilized to probe these two aspects of data properties. We showcase corresponding FRMSE and nMSE results in Fig. and Fig. 7
questions: What
For the first question, we escalate the size of training trajectories from 256 to 4,096. Overall, with the size of training data increasing, ID/OOD generalization capacity of PDE forecasting models elevate considerably, which is amenable to the scaling property between data size and model performance in scientific machine learning (Subramanian et al. 2023). Notably, for OOD fRMSE results, iMOOE trained size 512 rival FNO trained size 2,048. For OOD nMSE results, iMOOE trained on size 512 even outperforms FNO trained on size 4,096. indicates that the proposed PDE invariance learning can enhance the zero-shot OOD performance and data efficiency of ordinary FNO. Using 1,024 training trajectories for iMOOE can reach satisfactory zero-shot OOD results on DR dynamics compared to naive FNO.
For the second question, we keep the total number of training samples at 1,024 and alter the number of training environments from 4 to 512. The training data quantity in each environment is equal.
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We depict the distribution of ID and OOD results of each test sample in Fig. ((b). 7()). Overall, with diverse training domains, ie. when the number of training environments is up to 32, ID/OOD results of cach test trajectory can disperse domains smaller, and the average ID/OOD FRMSE much lower. This reveals that when data budget is limited, better data diversity can avoid overfitting limited training domains, and aid to find the fundamental PDE invariance principle by equalizing the risks across more diverse training environments. This coherent with the key claim foundational invariant learning literature (Ar- jovsky et al. 2019): with a sufficiently large number of diverse training environments, invariant risk minimization would elicit the invariant predictor.
compactly. In other words, the variance across test
ID
ID-FNO
0.0030
3.005
Mean
000+M00F
3.004
FRASSE
FRAISE
3.003
0.0015
3.002
2.001
3.000
0.0000
256
4096
16
120
512
Training size
Number of environment
(a) Scaling training data size.
(b) Fixed data collection budget.
Figure 6: Impact of training data properties ID/OOD fRMSE from two views: (a) Varying data size. (b) Varying data diversity under limited data budget.
10
0.10
0.20
J0HMOOe
Moan
BOD-FNO
0.08
600-MNX
2.15
0.06
0.04
3.05
1
0.02
3.00
0.00
36
64
256
512
Training data size
Number of environment
(a) Scaling training
(b) Fixed data collection budget.
Figure 7: Impact of training data properties on ID/OOD nMSE from two views: (a) Varying data
size. (b) Varying data diversity under limited data budget.
F.8 RUNTIME COMPARISON
We compare the runtime of both the PDE forecasting methods presented Table the commer- cial numerical solver Comsol (Multiphysies 1998) in Table 21 We can that the deep learning methods can lead to a nearly 225 x times speed-up on inferring the BG flow trajectories in contrast to the inner finite clement method in Comsol. It is hard for Comsol to converge when simulating the turbulent flow (i.e. the viscosity coefficient v BG small). At the same time, neural PDE methods modeling the real-world PDE systems.
obviate the need for complicated domain knowledge Besides, it is apparent that FNO-iMOOE indeed incurs extra computational burden on top of vanilla FNO, while its running speed is similar other OOD forecasting methods for PDE dynamics.
Table 21: Runtime comparison of different PDE dynamics simulation methods on BG data.
Methods
Comsol
ENO-M0OE
PNO
CAPE
VCNeF
DPOT
CNO
GEPS
CoDA
Inference Time 24. 8432.73
0.1 100.02s
0.05g0.002s
0.0650.002s
0.11
0.09:0.002s
0. 0.14 0.1 0.03s
0.07+0.0022
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F.9 IMPLEMENTATION DETAILS
We clarify nyperparameter settings for all baseline methods in Table 1 The fixed training setups include 32 training batch size and le initial rate for Adam optimizer.
CoDA. We train CoDA for 1500 epochs. The hidden dimension of shared hypernetwork and domain-specific 4-aayrr FNOs is 64. The weight of its L1 and L2 regularization on hypernetwork parameters le
CAPE. We train CAPE for 500 epochs. The widdening factor of its channel attention and width of 4-layer FNO backbone are 64. The weight of additional loss Leape is 8.3e 5.
CNO. We train CNO for 500 epochs. The channel multiplier of its UNet-shaped operator is 16. The hidden dimension and layer number of its bottleneck network is 128 and 4.
DPOT. We finetune the pretrained DPOT of tiny version for 500 epochs. The latent dimen- sion Fourier attention and FFN layer 512. The number of attention head is 4.
VCNeF. We train VCNeF for 500 epochs. The latent dimension and patch size of the linear transformer block is 64 and 16. The depth of modulation blocks is 4.
GEPS. We train GEPS for 1500 epochs. The width of domain-specific 4-layer FNO is 64 and code size of context vector is 16.
F.10
VISUALIZATION ON OOD FORECASTING RESULTS IMOOE
we visualize the forecasting outcomes of iMOOE on representative OOD physical environments of the five PDE dynamical systems.
In Fig. to 14
Step=1
3tep=3
Step=
Step=7
Step=9
Step=11
Target
Prediction
Figure 8: OOD forecast showcase on a DR scenario with D, = 0.0021, D, =
0.0113,13... 0.0109.
Step=1
Step=3
Step=
Step=7
Step=9
Step=11
Target
Prediction
Figure 9: OOD forecast showcase nigh-Reynold number NS scenario with 1.42c-4.
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Step=1
3tep=3
Step=5
Step=7
Step=9
Step=11
Target
Prediction
Figure 10: OOD forecast showcase on a low-Reynold number NS scenario with 1.200-3.
Step=1 1
Step=3
Step=5
Step=7
Step=9
Step=11 11
Target
Prediction
Figure 11: OOD forecast showcase on a high-Reynold number BG scenario with v 2.5e-3.
Step=1
Step=3
Step=
Step=7
Step=9
Step=11
Target
Prediction
Figure 12: OOD forecast showcase low-Reynold number BG scenario with 1.0e-1.
Step=1
Step=3
Step=5
Step=7
Step=9
Step=11
Target
Prediction
Figure 13: OOD forecast showcase SW scenario with initial radius.
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Step=1
Step=3
Step=5
Step=7
Step=9
Step=11
Target
Prediction
Figure 14: OOD forecast showcase on HC scenario with my 2.67, m2 12.66, my 2.74.
Step=1
Step=6
Step=11
Step=16
Step=21
Target
Prodiction
Figure 15: OOD forecast showcase on the x-axis velocity around Airfoil with unseen conditions.
Step=1
Step=6
Step=11
Step= 16
Step=21
Target
Prediction
Figure 16: 0OD forecast showcase the y-axis velocity around Airfoil with unseen conditions.
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Reviews:
[{"review_id": "oW6OsGjBti", "reviewer": "Reviewer_RXdg", "summary": "This work is motivated by invariant learning theory and addresses the challenge of out-of-distribution (OOD) prediction, covering both temporal and parametric extrapolation tasks. The key idea is that partial differential equations (PDEs) possess two fundamental types of invariance:\n* Operator invariance: individual physical operators (e.g., diffusion, advection, reaction) remain consistent across domains.\n* Compositionality invariance: the relationships between these operators and external factors such as physical parameters and forcing terms remain fixed for a given PDE.\nTo capture these invariances, the authors propose iMOOE (Invariant Mixture of Operator Experts). The method first decomposes a PDE into a set of compositional operators, then uses a Mixture of Operator Experts architecture to model both the invariant operators and their compositional relationships.\nThe effectiveness of iMOOE is demonstrated across five canonical PDE systems, including diverse parametric and temporal OOD scenarios, as well as a real-world Sea Surface Temperature (SST) dataset. Across all tests, iMOOE consistently outperforms baseline approaches (such as CoDA, GEPS, CAPE, CNO, DPOT, and VCNeF), achieving state-of-the-art performance in challenging OOD prediction tasks.", "questions": "1. How does iMOOE perform for irregular spatial domains or unstructured meshes? Can the operator decomposition and masking strategy be naturally extended to such cases?\n2. Can the authors comment on the generalization between dimensions — for example, if trained on 2D heat conduction, how well would the approach extrapolate to 3D?\n3. Beyond parametric shifts, can iMOOE handle structurally different PDEs (e.g., transitioning from Burgers to Navier–Stokes), or does operator invariance only hold within a single PDE family?\n4. How sensitive is the method to the diversity of training domains? Is there a minimum level of variation required to achieve robust generalization?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper presents a framework for improving OOD generalization in PDE simulation, motivated by invariant learning theory.\n2. Several test cases are provided, including both temporal and parametric OOD settings, validating the method’s potential on challenging extrapolation problems.\n3. The paper includes comprehensive experiments, systematic comparisons with strong baselines, and sensitivity analyses (e.g., number of experts, loss weighting, domain partitioning).", "weaknesses": "1. The method assumes that PDEs can be decomposed into known invariant operators. In cases where the governing equations are unknown or highly complex, this assumption may not hold, reducing the method’s applicability.\n2. The paper claims “zero-shot OOD generalization”, but the evaluated OOD shifts are primarily controlled parametric variations. It is unclear whether this corresponds to zero-shot generalization to entirely new physical equations.\n3. The method’s reliance on multiple neural operator experts increases computational cost, especially for high-dimensional PDEs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GeXD2ANATt", "reviewer": "Reviewer_adhf", "summary": "This paper addresses the challenge of out-of-distribution (OOD) generalization in PDE dynamics forecasting. The paper proposes iMOOE, consisting of: \n- A Mixture of Operator Experts (MOOE) architecture, where parallel neural operator experts model distinct physical processes, and a fusion network captures compositional relationships.\n- A frequency-enriched invariant learning objective, combining prediction, risk equality, and frequency regularization losses to promote invariant representations across environments.\nExtensive experiments on five PDE systems (Diffusion-Reaction, Navier–Stokes, Burgers, Shallow-Water, Heat-Conduction) show that iMOOE improves ID and OOD generalization over baselines.", "questions": "See weaknesses", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Architectural Innovation:\nThe MOOE architecture aligns with PDE operator decomposition, offering modularity and interpretability. Its parallel design is more efficient than serial operator splitting.\n\n- Robust Empirical Validation:\nThe experiments are comprehensive—covering both synthetic PDE benchmarks and a real-world application (SST).", "weaknesses": "- Limited Diversity of PDE Systems:\nAlthough the paper evaluates multiple canonical PDEs, they are all relatively low-dimensional and well-behaved (2D systems with periodic boundaries). The scalability to irregular geometries or 3D turbulent systems is unclear.\n\n- Theoretical Rigor of Invariance Definition:\nThe proposed “two-level PDE invariance” is intuitive but lacks rigorous formal grounding. The connection between the invariance principle and the optimization objective could be better substantiated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "b1oCU7SOzL", "reviewer": "Reviewer_K9mK", "summary": "This paper proposes iMOOE, a physics-guided invariant learning framework for zero-shot out-of-distribution (OOD) generalization in PDE dynamics forecasting. The key idea is to exploit two levels of PDE invariance—operator invariance and compositional invariance—through a Mixture of Operator Experts (MOOE) architecture and a frequency-enriched invariant learning objective. The authors show improved ID and OOD performance across several simulated PDE systems (Diffusion–Reaction, Navier–Stokes, Burgers, Shallow-Water, and Heat-Conduction) compared with recent meta-learning and parameter-conditioned models.", "questions": "1. Can the authors provide more formal or causal justification for the “two-level invariance” concept?\n\n2. How sensitive is the method to the way training environments are defined (by parameters vs. time steps)?\n\n3. How does the model perform on noisy or irregular real-world PDE data, such as CFD or weather simulations?\n\n4. Does the frequency loss harm stability when data are non-periodic or contain sensor noise?\n\n5. What is the computational overhead of iMOOE compared to a standard FNO?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The paper tackles an important and difficult problem “zero-shot OOD generalization in PDE dynamics” and does so with a physically motivated approach.\n\n2. The proposed Mixture of Operator Experts (MOOE) architecture is intuitive and nicely aligned with operator-splitting ideas in classical PDE solvers.\n\n3. The frequency-augmented loss and risk-equality objective are interesting design choices that improve empirical robustness.\n\n4. Experiments on multiple PDE benchmarks show clear performance gains over meta-learning and conditioning-based methods.", "weaknesses": "1. The “two-level PDE invariance” principle feels more heuristic than theoretically grounded — there’s no rigorous justification or proof that these invariances hold across PDE families.\n\n2. All experiments are on synthetic, noise-free datasets with simple geometries and periodic boundaries. No real or noisy data are tested, so generalization claims are not fully convincing.\n\n3. The frequency-based regularization might amplify noise in realistic cases, but this is not examined.\n\n4. Comparisons with baselines may not be entirely fair since meta-learning models are designed for few-shot adaptation rather than strict zero-shot settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0edzN8JxzP", "reviewer": "Reviewer_YiSh", "summary": "This paper tackles zero-shot OOD generalization for PDE forecasting with **iMOOE**, a framework built on a **\"two-fold PDE invariance principle\"**. This principle posits that core physical operators and their compositional structure remain invariant despite parameter shifts. The framework captures this using an **invariance-aligned Mixture of Operator Expert (MOOE) architecture**  and a **frequency-enriched invariant learning objective**. This objective enforces risk equality while also using a novel regularizer to **mitigate the spectral bias** of neural operators, forcing them to learn essential high-frequency features. Experiments show this \"plug-and-play\" framework significantly boosts zero-shot OOD performance for various neural operators.", "questions": "1.  The paper's premise is that prior OOD methods fail *because* they miss the \"fundamental invariance principle.\" Is there direct evidence to support this causal claim about *why* baselines fail, beyond iMOOE's own success?\n\n2.  To validate the core hypothesis, can you provide an interpretability analysis of *what* the individual experts $\\sigma_i$? For example, in the DR system, do they measurably specialize into distinct diffusion and reaction operators?\n\n3.  Could you clarify the exact implementation of the derivative selection masks $m_i$? Are they learnable parameter vectors for each expert, or the output of a separate routing/gating network?\n\n4.  To isolate the benefit of the MOOE *structure* from simply having more parameters, could you add a baseline comparison against a single, monolithic operator with an equivalent total parameter count to the $K$-expert iMOOE?\n\n5.  While iMOOE uses dense experts, does it face training challenges analogous to sparse LLM MoEs, such as expert collapse or redundancy (especially as $K$ increases)? How effective is $\\mathcal{L}_{mask}$ at ensuring true specialization versus mere diversity?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper tackles the highly significant and challenging problem of zero-shot OOD generalization for PDE dynamics. It accurately pinpoints the failure of prior methods: they do not capture the underlying physical invariance required for true zero-shot generalization without test-time adaptation.\n\n* The core originality lies in proposing a \"two-fold PDE invariance principle\" (operator and compositionality invariance). This provides a structured, physics-based hypothesis that reframes the problem from finding abstract features to learning specific, invariant physical components.\n\n* The framework is technically novel and significant in two ways. First, its \"plug-and-play\" nature makes it a broadly applicable tool for enhancing various existing neural operators. Second, it originally introduces a **frequency-enrichment loss** ($\\mathcal{L}_{freq}$) to mitigate the spectral bias of neural operators, which is a critical and novel step for learning the complete physical invariance.\n\n* The paper's claims are supported by comprehensive experiments across simulated PDEs and a real-world dataset, using a strict OOD setup with non-overlapping parameters. The results consistently show SOTA performance. Crucially, the insightful ablation studies (e.g., on expert number $K$ and the $\\mathcal{L}_{freq}$ loss) effectively validate that each core component of the framework contributes to the final success.", "weaknesses": "* The paper's central premise—that prior methods fail *because* they miss the \"fundamental invariance principle\"—is asserted without direct evidence. This creates a correlation-not-causation gap, as the paper only proves its own method works, not *why* others fail.\n\n* The paper fails to validate its core hypothesis with interpretability. There is no analysis or visualization of *what* the specialized experts ($\\sigma_i$) actually learn, nor any theoretical analysis of the decomposition's properties (e.g., completeness, uniqueness).\n\n* The mechanism for learning the derivative selection masks ($m_i$) is ambiguous. It is unclear whether these masks are learnable parameters or the output of a separate network, which harms reproducibility.\n\n* A crucial baseline is missing: a monolithic neural operator with an equivalent parameter count to the final $K$-expert MOOE model. Without this control, it is unclear if the performance gain comes from the principled MOOE *structure* or simply from increased model capacity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)