A Plug-and-Play Query Synthesis Active Learning Framework for Neural PDE Solvers
Authors: Zhiyuan Wang
Jinwoo Go
Byung-Jun Yoon
Nathan Urban
Xiaoning Qian
Venue: NeurIPS 2025 poster ()
Keywords: active learning
neural PDE solver
query synthesis
expected information gain
Primary area: machine_learning_for_sciences

Abstract: In recent developments in scientific machine learning (SciML), neural surrogate solvers for partial differential equations (PDEs) have become powerful tools for accelerating scientific computation for various science and engineering applications. However, training neural PDE solvers often demands a large amount of high-fidelity PDE simulation data, which are expensive to generate. Active learning (AL) offers a promising solution by adaptively selecting training data from the PDE settings--including parameters, initial and boundary conditions--that are expected to be most informative to help reduce this data burden. In this work, we introduce PaPQS, a Plug-and-Play Query Synthesis AL framework that synthesizes informative PDE settings directly in the continuous design space. PaPQS optimizes the Expected Information Gain (EIG) while encouraging batch diversity, enabling model-aware exploration of the design space via backpropagation through the neural PDE solution trajectories. The framework is applicable to general PDE systems and surrogate architectures, and can be seamlessly integrated with existing AL strategies. Extensive experiments across different PDE systems demonstrate that our AL framework, PaPQS, consistently improves sample efficiency over existing AL baselines.

A Plug-and-Play Query Synthesis Active Learning Framework for Neural PDE Solvers
Zhiyuan Wang', Jinwoo Go², Byung-Jun Yoon!:2, Nathan Urban?, Xiaoning Qian 1:2,3
Department of Electrical & Computer Engineering, Texas A&M University, College Station, TX Computing & Data Sciences, Brookhaven National Laboratory, Upton, NY DDeaarmment of Computer Science & Engineering, Texas A&M University, College Station, TX
Abstract
In recent developments in scientific machine learning (SciML), neural surrogate solvers for partial differential equations (PDEs) have become powerful tools for accelerating scientific computation for various science and engineering applica- tions. However, training neural PDE solvers often demands a large amount high-fidelity PDE simulation data, which are expensive to generate, Active learning (AL) offers promising solution by adaptively selecting training data from the PDE settings-including parameters, initial and boundary conditions that are expected to be most informative help reduce this data burden. In this work, we introduce PaPQS, Plug-
Plug-and-Plaa and-Play Query Synthesis AL framework that synthesizes infor- mative PDE settings directly in the continuous design space. PaPQS optimizes the Expected Information Gain (EIG) while encouraging batch diversity, enabling model-aware exploration of the design space via backpropagation through the neu- ral PDE solution trajectories. The framework is applicable to general PDE systems and surrogate architectures, and can be seamlessly integrated with existing AL strategies. Extensive experiments across different PDE systems demonstrate that our AL framework, PaPQS, consistently improves sample efficiency over cxisting AL baselines.
Introduction
Partial differential equations (PDEs) are fundamental tools for modeling a wide range physical phenomena in science and engineering. Traditional numerical solvers, such finite difference and finite element methods [1], provide accurate solutions but often entail significant computational costs, especially when high spatial and temporal resolutions required. The high computational cost becomes critical bottleneck applications involving fast or extensive evaluations, especially for consequent uncertainty quantification, optimization, and decision- making 3].
Neural PDE solvers have recently been developed as efficient surrogates alleviate this cost, enabling accelerated uncertainty analysis and decision making 4 5 6 7 Unlike typical supervised learning tasks, training neural PDE surrogates must account for spatiotemporal complexity and sensitivity to initial and boundary conditions, particularly in complex systems with chaotic or highly nonlinear dynamies. Therefore, randomly simulating many high-fidelity trajectories is impractical, given its prohibitive time and cost. These challenges motivate the development of new active learning (AL) methods tied to PDE systems, which seek to improve data efficiency by selectively querying the most informative PDE settings rather than relying exhaustive data. Recent studies have evaluated AL for neural PDE surrogates, aiming to reduce the number of expensive simulations while maintaining neural surrogate model accuracy [8]
13]
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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One representative class is adaptive active sampling approaches [5] which prioritize train- ing points based on PDE residuals estimated over continuous coordinates. However, these methods in- surrogates such as physics-informed neural network (PINN) [7 I8
herently rely on coordinate-base and cannot be readily applied to widely used grid-based neural surrogates such as U-Net 7 19].
Another class typically relies on pool-based selection strategies [8 (3, 21 22], which are con- strained by the limited coverage of candidate pools and lack flexibility to explore the continuous design space. Moreover, since the pool is agnostic to the neural surrogates, these methods cannot dynamically respond to training progress or the model uncertainties, limiting the ability to refine queries. Furthermore, predictive variance is commonly used as the acquisition function in scientific modeling to evaluate data informativeness based on the model's ' epistemic uncertainty [8] 12 21]. But maximizing the variance could lead to chaotic PDE solutions that distort acquisition signals, particularly under recurrent prediction settings [5] 7 23 Meanwhile, Teature-based acquisition methods 1 24 25 emphasize input diversity but do not account for the relationship between PDE settings and model parameters. While recent pool-based methods have explored combining uncertainty- and diversity-based criteria [26] 27), they often require careful designs and remain limited to discrete candidate sets. There is a pressing need for a general mechanism that can flexibly plug into various acquisition strategies and enhance them by enabling continuous and adaptive query synthesis for the neural PDE solver problem.
To address these limitations, we propose a Plug -adg--an--ua Play Query Synthesis (PaPQS) active learning framework for neural PDE solvers that directly synthesizes informative PDE settings in the continuous design space. Rather than depending of candidate settings by ascending the expected information gain (FIG) an acquisition function that directly interacts with the neural PDE solver and its parameters. By explicitly targeting the reduction of the posterior uncertainty, our framework dynamically explores high-informative regions while avoiding instability-prone areas that can mislead traditional uncertainty-based methods. We also introduce policy function that incorporates an entropy-based regularization term to balance informativeness and batch diversity, encouraging wide coverage of the continuous search space and mitigating mode collapse. Importantly, PaPQS is designed in plug-in style and can be readily combined with well-studied active learning strategies to further boost performance. This makes it versatile framework that improves both the resolution and flexibility of sample acquisition, ultimately leading to more efficient training of neural PDE surrogates.
fixed candidate pool, our framework optimizes batch
In summary, our main contributions include:
We propose PaPQS that directly synthesizes informative PDE settings in the continuous search space, overcoming the rigidity of pool-based strategies. It considers both EIG and batch diversity, enabling adaptive exploration in the continuous space.
We introduce an efficient critic-based approach to estimate EIG and its gradient with respect the parametrized search space, allowing batch optimization ascend informativeness. Our framework is extensible to general PDE systems and neural PDE surrogate architectures.
Extensive experiments demonstrate that PaPQS generalizes well across diverse PDE systems and surrogate architectures, achieving consistent gains in sample efficiency and showing strong compatibility when combined with existing active learning strategies.
2 Background
2.1 Neural PDE surrogates
We focus on PDEs over a spatial domain X' and temporal domain [0, 7]. with solution u(t, x) E NRN. where N₂ denotes the number of physical variables or channels. Without loss of generality, the PDE formula can be written as follows:
u,u = F(A, L u, Uz Sozu,...), (1,x) E (0, T x X
u x)
n (x), x X: B[ul(t,)==0. (t,x) E (0, x ax
(2)
Here, B is the boundary condition. In this paper, we restrict our attention to periodic boundary conditions- - the most commonly used setting in SciML studies 28 for simplicity and in line with prior benchmarks [3] X E TR. denotes the PDE parameters (e.g., viscosity in the Navier-Stokes equations), and u represents the initial state of the system. We define the initial condition (IC),
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2/ = = X), which includes both initial state and PDE parameters in this paper, could be drawn from a target input distribution Pr(+) pr(u")pr(A) One popular choice for IC generator that draws u from a superposition of sinusoidal functions with random parameters (28) and samples À
from uniform distribution Uniformaa,h) b). In this paper, we use this method to generate ICs and select or synthesize ICs via their generative parameters.
The ground-truth PDE solution traditionally computed using a numerical solver that iteratively propagates the state forward in time, which is often computation-demandin and time-consuming, To accelerate such PDE simulations and enable real-time decision-makee neural surrogates have emerged as efficient alternatives 7 I8 29 and achieved success in various domains [8 30 3T] In this paper, we discuss the neural PDE solver with an autoregressive setting that predicts the next state as û(t + At. -)- = Ma (ū(t. A)). The training data consists of input-output pairs (3b, u), where ub from the IC generator and u is the trajectory simulated by a numerical solver. The model is trained by minimizing the root mean squared error (RMSE):
LRMSE( (u, û) N.
[u((),,z)) ú(L,, 2,)
where i is the predicted solution from the neural surrogate model, N₂ and N₂ denote the resolution of the discretized temporal and spatial domains, respectively. While this autoregressive setting serves as example, note that our proposed framework broadly applies other types of neural PDE surrogates.
2.2 Expected information gain
The Expected Information Gain (EIG) is principled acquisition criterion widely used in Bayesian optimal experimental design (BOED), where the goal to select experimental conditions that maximize the information gained about unknown parameters [19]. With the IC input tuple vb, neural surrogate model parameters 0, and observed solution fields u, is formally defined as:
p(u 0, +))
ETG( A (()())p())e))) log
p(u
where p(0) the prior distribution over surrogate parameters 6, p(u(0, 2b) the likelihood function given a data point at vb. This expression captures the expected reduction in posterior uncertainty and forms the basis of information- theoretic acquisition in BOED and active learning.
2.3 Active learning for neural PDE solvers
We now provide formal problem definition of our active learning framework for neural PDE solvers. The goal is train more accurate neural PDE solver Ma using fewer labeled data points. Let Suain denote an initial training dataset, and MM be the model trained on it. At each AL iteration, we aim sample batches informative ICs, Su.. query their labels (i.c., solve the numerical solvers), and augment the training set. This procedure is expected to yield faster convergence compared to randomly sampled ICs. We then retrain Ma on the updated dataset Strain U Sür, and evaluate performance on held-out test set with ICs drawn from the target distribution (tb).
A standard practice in active learning for neural PDE solvers is the pool-based methods 8 13 21 [22] 32), where a candidate pool Spool is randomly sampled prior to active learning. At each predefined acquisition
iteration, subset of informative ICs Sir is selected from Spoot according function, labeled using numerical solvers, and added to the training set. The selected samples are subsequently removed from Spoot- For the acquisition function considering interactions between model and data, the majority of existing studies rely on the predictive variance estimated either by model ensembles
13
32 or inferred from the predictive distribution of probabilistic models [8]
3 Method
We now introduce our proposed AL framework PaPQS for neural PDE solvers, illustrated in Fig. Instead of selecting from a fixed candidate pool, PaPQS operates directly in the continuous space of zb = (u", A), synthesizing informative ICs by ascending the policy function. This design supports
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IC generator for target PDE
IG random active generation strategies
reg www.eeeeeeeeee end regurrge beleetle
PaPQS iteration for IC synthesis
Random sampling
Feature- based active learning
Uncertainty-
Numerical PDE solver
active learning
Others
Gradient- bastdddaaddd-- operator
Gradient calculation update by (5s. (13), (14)
Initial IC batch S/3)
EIG estimation
Training data
by (q.(11)
At not last update stop
Train Neural
Neural PDE
Refined
Retain or discard by poloyy score
Policy score by Eq.(9)
PDE solvers
solvers
SWAG
IC batch S
Figure 1: Overview of our PaPQS framework, including (a) the active learning pipeline for neural PDE solvers guided by PaPQS (red part) with (b) iterations of the PaPQS procedure (green part).
higher-resolution surrogate training, greater sampling flexibility, and more efficient data acquisition. Moreover, we aim for plug-and and-play design that can be readily applied to refine candidate ICs from existing AL strategies.
Based on this idea, we produce batches of ICs, Seach { Muug J. with high informativeness which is evaluated by the policy function and label them for neural PDE surrogate's 's training at each AL iteration. Starting from an initial batch Span the IC synthesis process can be formulated as an optimization objective:
Shutch arg max T (Stacch). Sench
(4)
where (--) is the policy function that quantifies the informativeness of a given batch through an acquisition measure and a diversity term. our framework, this optimization performed via a gradient-basse based update procedure with explicit control of batch diversity as described in Sec. The acquisition measure in T iT-)) is defined as the EIG, whose estimation and gradient computation are detailed in Sec.
3.1 Gradient- operator
The optimization can be further designed to jointly capture individual informativeness and overall batch diversity. The synthesis of the next IC is guided towards regions of higher acquisition function scores, thereby improving the utility of individual new samples to acquire. Simultaneously, we impose a diversity-aware tradeoff to prevent the batch from collapsing into redundant regions, ensuring that the selected samples remain representative and diverse. This joint consideration enables to synthesize training batches that are both informative and well-spread in the design space.
Specifically, we boost individual IC samples ub via gradient ascent on the acquisition function using Adam optimizer (33,, the single update step is
(m+1)
(5)
Gin +e
where A denotes the acquisition function that directly determines the gradient direction for ICs refinement, instantiated as the EIG in this work (see Sec. 3.2,, 6(m) = T- ( (n-1))+(( +) winh A(xb
m (m-1) + V d (mn))AAA A(tb are moment terms, a is the step size determining the and (m) T-8, magnitude of IC movement and n E [D NNtepp denotes the n-th update step.
9.
Eq. (5) facilitates an adaptive and gradient- based search procedure for active query synthesis in continuous design space, guiding each sample toward more informative regions while maintaining numerical stability through moment- based updates. In particular, larger update step Naup promotes broader exploration, allowing the optimizer to escape suboptimal basins and identify regions with higher expected information gain, while smaller step emphasize local exploitation, refining samples within high-EIG neighborhoods and maintaining batch-level diversity. This adaptive mechanism provides principled trade-off between exploration and exploitation, crucial for efficient active learning on IC query synthesis.

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To further enhance batch quality and ensure that the transformed samples are both informative and well-distributed, we introduce an entropy- based diversity regularization term, which is expressed by
H. (n)) i Strain batch) = log (
(n)
Suainn batch)
(N (es + 1).
(6)
d we?0")
Here, H-) could be regarded as an batch entropy estimator based on the k-nearest-neighhor and Kozachenko-Leonee Leonenko entropy [3 35], with Dx( (zb Swain Stacc) is the distance between v/(")
and its k-nearest neighbor in the space of Strain U S N (D,((o!"))) denotes the number of training
points whose distance to (b(") is less than (x(+b?), d dos(")) is the dimensions of w/"h, and F (.) is the digamma function. This entropy term penalizes samples either densely clustered or located in regions already sufficiently represented by labeled data.
The policy function zz-) now consists of an acquisition term A(.), which indicates informativeness, and diversity regularization term H((-), which measures spatial diversity. At each update step, calculated by
A(tb! (m) + H((b!"); Smann S
where y is a hyperparameter balancing each term. Finally, we use this policy score to determine whether to retain each updated sample, i.e.,
26 (n+1) if (Tb(((t))) T(b("))) -(bb))) 7
(8)
otherwise,
where 77 is threshold that filters out non-contributive updates.
3.2 ElG-based acquisition function and gradient estimation
Previous works on active learning for neural PDE solvers often adopt predictive variance - typically estimated from ensembles 13 21 or probabilistic models g as the acquisition criterion. However, predictive variance can yield misleading acquisition signals neural PDE settings for two reasons. First, many PDEs exhibit chaotic or highly sensitive behavior (e.g., the KS equation), where predictive variance may spike in response to small perturbations, without indicating true informativeness. Second, neural PDE solvers often adopt recurrent prediction schemes, where small prediction errors accumulate over time, further amplifying variance and distorting the acquisition signal [5] 7 These effects drive the synthesis toward uncertain yet unstable and uninformative regions of the IC space, ultimately degrading sample efficiency and model performance.
We adopt EIG 3 as our acquisition function as it effectively quantifies the expected uncertainty reduction over model parameters on potential observations, justified in Appendix A..
Claim 1. Not all uncertainty is epistemically informative. It is therefore crucial to distinguish whether the observed uncertainty arises from a lack of knowledge about the model parameters (epistemic) from intrinsic trajectory instability (aleatoric or chaotic) that cannot be reduced through learning 136]. Suppose predictive variance is used as the acquisition function. Then, preferred regions may reflect high prediction uncertainty (formulated as Vare=p(o) E/u 0, +))), which can dominated by chaotic sensitivity rather than epistemic uncertainty, and thus fail to provide useful information about 9. In contrast, EIG explicitly quantifies the informativeness of vb with respect to updating the model parameters by minimizing posterior uncertainty, thereby prioritizing epistemically informative regions and supporting robust. data- data-efficient neural PDE surrogate learning
However, since most neural PDE surrogates are implicit models without tractable likelihoods, we instead estimate the FIG by leveraging equivalence to mutual information (MI). Specifically, we employ the MINE-f estimator lower bound mutual information [39 40 This can be expressed as:
Estimating EIG requires access to the likelihood function p(u(0, (b)
37 38
EIG($) - MI(u) 0) = (KL(((( u) p(0)p(u))
> Ep(e.u) [To w) e - Be (o
where (0, u) a neural critic function parameterized by with neural solver parameters 0 and predictions as input.
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At each AL iteration, parameters ø could be optimized by maximizing ( over the joint samples (0,, u;) p(0)p(u 0, ab) and independent samples 8, ~ p(0) and u, ~ p(u f Φ),, where f, is another sample from p(0) with & i. To draw posterior parameter samples after training with the currently updated p(f), an easy and popular way is to train an ensemble of surrogate models, where each model corresponds to different approximate posterior sample [41]. Instead, we adopt more efficient method, SWA-Gaussian (SWAG) [42], since it avoids the time-consuming ensemble training process, which is important for typical computationally constrained AL settings.
Using SWAG, the posterior samples 8, are generated as follows:
1
0. = OswA +
V2 ding +
/2(K - D22,1, 21, ((0, la,), 22,4 (00 I Nswa ). (10)
Here, we denote + = (0x keEI the trajectory of parameters
the last NSWA neural solver training epochs. Then ASWA is the mean value of O. Edag is the variance calculated by Eding
w- D is the deviation matrix comprised of columns D2 = 02 0sWA- For computational efficiency and to focus uncertainty estimation on the output space, we apply this sampling procedure soiely to the output layer of the neural surrogate, keeping all preceding feature layers fixed, justified by the following proposition.
diag( News - 82
Nsws
Proposition 1. Let nerual PDE surrogate model parameters be partitioned as 0 = (0jeat: oour), where an is fixed and only ouw is sampled. Assume that uncertainty over surrogate predictions u is dominated by the output layer parameters Oow, i.e., p(u 0, v)) = p(u Oon, 16). Then, the expected information gain (EIG) can be approximated by sampling only Oonn. i.e., (IG(()) = [(0; u up)
10mii 1p). valid lower bound on the EIG can be estimated by sampling only Oow given by:
(+) 2 E)((.... (Oour, u)] -e e -Ep(0aa)p(())
(e [T((0w)))))
(11)
Proof is provided in Appendix cost based on (10), and can be computed in (NNwwdd)) time. We note that this approach could also be replaced by other efficient posterior approximation techniques, depending on practical considerations.
A.2 Posterior sampling can be performed without additional training
As for the sample of u, it corresponds to the predictions derived by the neural PDE solver, given a sample pair (0,, vb)). Here, we use the ICs +b E S as inputs to get predicted solutions [u(t, vp; ( and form joint samples as (0), ult,, 0.)). Based on this setting, the training dataset for optimizing MINE-f is constructed to align the distribution of ICs to be updated, en- abling it to effectively estimate the EIG during the gradient ascent process. To obtain the independent samples, we perform a random batch permutation over the set of joint samples.
With the MINE-f serving lower-bound estimator, we can now compute the gradient described in Eq. (5) Besides, we can also estimate the gradient with respect to the critic parameters o, i.e., Va.(w))")) We therefore adopt an amortized update strategy (3 that jointly updates both the ICs dp and the critic network é which improves the accuracy and stability of EIG estimation throughout the gradient ascent process. The gradient at the n-th update step is then calculated as follows:
18(n) 1 A(2b(n))
= V
mo(") (Bp(e.n) To(O, u)] - -e- "Bp(o)o(()) (Bpte)n(uy/e22000..)) (626(0.00)
1
N₂
e-
Nuua
(1
(o,,,, (t;t,b;;0,))
N
(n)
N Nbacc
1
V To (0,, ((:))) V ut,
(12)
N
Nonnhh
V Val(ab(
V (o u, u.(4.b,;))))
Vse 7.0,
(13)
N
Nhatch
For computational efficiency, we compute gradients only at three prediction steps (1, [N,/2], NN N.) - and average them to update the ICs. The complete procedure of our framework is provided in Appendix B

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4 Experiments
To discuss the efficacy and efficiency of the proposed PaPQS, we focus on the following research questions empirically: (1) Performance. Does PaPQS work? Fig. 2 compares PaPQS with -f the-art baselines using four commonlhy tested PDEs; (2) Generalizability. Does it support versatile verifies the adaptability of PaPQS across different models; (3) Reusability. Does PDE data synthesized by one architecture improve the training performance of other architectures? Fig. 3b presents a case under the arshtttrcee setting; (4) Efficiency. Does it achieve comparable or superior performance with reduced training time? Figs. [c and [3d] present an 'accuracy vs. wall-clock time' analysis to demonstrate the data efficiency of the proposed PaPQS. (5) Component. How do components work? Figs. 4a and 44b conduct ablation studies to dissect the influence of each mechanism. (6) Sensitivity. What is the impact of varying parameters?
neural PDE surrogate architectures? veriifis
Fig. 4e offers a sensitivity analysis of the key hyperparameters the number of gradient update steps.
Additional empirical results and case studies are provided in Appendix E
4.1 Problem Setup
PDEs, neural surrogates, and baselines. We consider four PDEs with periodic boundary conditions as studied in recent AL4PDE benchmark [13 (1) Burgers' equation with vis- cosity [28]: 0,u + ud,u u = (4) uz (2) Kuramoto-Sivashinsky (KS) equation [T: 0,u + ud u + 0z-u + v0
0; (3) Combined Equation (CE) [43 with the forcing term 5 = 0:
0,u + 0z (au² 30,u+ +00zu)) = 0: (4) Compressible Savier-Stoker equation (28 in 2D space Vv) (2D-CNS): 0.p- + V. (pv) = 0, 08.vv vV.v+ = -Vp + nyv + (S+ n/33 V(V .v), v a') 0. We include three neural PDE surro- gate solvers: (1) a modern U-Net (5) for PDE modeling; (2) SineNet 77] an improved U-Net that mitigates feature misalignment by evolving high-resolution features through multiple sequential waves and achieves the -fate-of-thh-ar performance; (3) the Fourier Neural Operator 44] (FNO). The hyperparameters used for IC generation and neural PDE solvers are aligned with those in
0, pv2 /2) + V [p+++
AL4PDE 13, with full details provided in Appendix C
We compare our proposed AL framework with the following baselines: (1) random sampling and Latin Hypercube sampling (LHS) as static Design of Experiment (DoE) baselines; (2) pool-based
AL strategies with predictive variance of an ensemble of models as the acquisition function. We select samples by Top-K and stochastic batch active learning (55 (SBAL) methods; (3) pool-based
strategies with the feature-based input selection, including CoreSet 24 and LCMD (25: (4)
hybrid strategy that first aggregates features using LCMD or Core-Set, and then applies BAIT 46 to
minimize the average posterior predictive variance 25
over the aggregated features.
Implementation details. We adopt the same training settings for the neural PDE solvers as those used in the AL4PDE [13 benchmark. For the hyperparameters in the gradient-based operator (cf. Eqs. (5)-(B)) we perform ten gradient ascent steps for each batch of ICs, using step size of a = 0.01 scaled by the range of each parameter's 's range. We set the balance hyperparameter Y to 0.5, and the filter threshold 17 to the median of the policy score computed in each batch. For EIG estimation, we record the parameter trajectory of the last 30 training epochs for the SWAG calculation in (IO). As for the architecture of the neural critic function T' (0, u), we employ a convolutional module to reduce the dimension of input and u, and then fuse them by a 3-layer MLP. Detailed training schedule and layer information are provided in Appendix D
4.2 Performance comparison and model analysis
Comparison of AL methods. We report the average aot--man-sqqarreeeee (RMSE) and 95% confidence interval in Fig. 2 Tables El E4 Results show that our PaPQS consistently improves model performance over baseline AL strategies for neural PDE surrogates. Besides, combining PaPQS with existing AL methods (e.g.. SBAL and LCMD) outperforms their stand-alone counterparts. For example for the Burgers' equations, our method reduces the average RMSE across iterations by 48.7%, 15.3%, and 7.4% when applied to candidate ICs obtained by random sampling, SBAL, and LCMD methods. This indicates that PaPQS provides the most benefit when paired with less informed strategies such as random sampling, with diminishing but consistent gains when combined with stronger baselines. This validates its strength as general-purpose refinement module that enhances sample informativeness across diverse acquisition methods. The varying gains across
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Figure 2: Error of U-Net over the number of trajectories in the training set. The shaded area represents the 95% confidence interval of the mean estimation calculated over multiple seeds.
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Figure 3: (a) RMSE on 2D-CNS equation: solid and dashed lines denote the random and ran- dom+PaPQS strategies. (b) RMSE on Burgers' equation: U-Net is used as the evaluation model, with either U-Net or FNO as the acquisition model. (c) RMSE on Burgers' equation over time. (d) Runtime of each part of our PaPQS in the "temporal behavior" " example.
PDEs can be attributed to differences in temporal and spatial complexity. For 2D-CNS equations, the improvement is moderate, which may be due to the system's 's complex, multi-variate coupling that makes informative IC optimization more challenging. Nevertheless, our method consistently provides stable performance improvement. With the results on Burgers' equations, characterized by simpler dynamics, showing pronounced benefits, the adaptability of our PaPQS framework has been demonstrated across diverse regimes including all these four PDE systems. We further evaluate our framework by changing the neural surrogate architecture. As shown in Fig PaPQS steadily reduces RMSE across all tested architectures, demonstrating its robustness and versatility.
Data reusability. We evaluate data reusability by testing whether a dataset synthesized using one neural surrogate architecture can benefit the training of another. Specifically, we train an FNO model to estimate the EIG and corresponding gradients, while retaining U-Net the evaluation model. Results are shown in Fig. 3b Both U-Net and FNO-guided synthesis significantly outperform random sampling in downstream evaluation, even when the evaluation model (U-Net) differs from the model used for acquisition (FNO). This highlights the strong reusability and cross-model generalization of our IC synthesis, demonstrating the plug-and -ad---dd-ppp play nature of our framework.
Temporal behavior. We evaluate the temporal behavior by tracking model performance over data generation time, which, for PaPQS, includes the total time consumption on training an acquisition model for EIG and related gradient estimations, training the neural critic function, gradient ascent, and simulation. Here, we estimate the acquisition function with an FNO with 30 training epochs, including 10 SWAG epochs for uncertainty estimation, and evaluate the performance with a U-Net model. Results in Fig. 3d indicate that PaPQS yields substantially lower RMSE than random sampling for the same data generation time budget. Although PaPQS introduces additional computations, these components account for less than 1/15 of the total runtime compared to simulation, as shown in Fig.3d The enhanced IC quality resulting from PaPQS leads to faster surrogate training convergence, underscoring its strong practical efficiency.
Different acquisition functions. We assess the impact of the acquisition function used in gradient ascent updates. While PaPQS employs EIG by default, we substitute it with predictive variance
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Figure (a) RMSE and runtime for updating a batch of ICs on the Burgers' equation under different acquisition functions. (b) RMSE and queried PDE parameter (viscosity) distributions with and without diversity regularization on the Burgers' equation. (c) RMSE on the Burgers' and KS equations under different update step settings.
estimated from an ensemble of neural PDE solvers. The results are shown Fig. 4a Overall, both acquisition functions achieve comparable performance across all dataset sizes, consistently outperforming the random baseline. Notably, in some iterations, the predictive variance method exhibits slightly wider confidence intervals, suggesting may introduce instability in AL.. More importantly, our EIG method offers significantly lower computational cost. This stems from the gradient structure: EIG estimation PaPQS relies contrastive formulation where gradients from negative samples can be safely truncated, since their trajectories are independent of the target IC (cf. Eq. [2)). In contrast, predictive variance requires multiple forward and backward passes for the same IC under different parameters, all of which contribute non-zero gradients. Consequently, optimizing variance incurs higher memory and runtime overhead, making EIG more efficient choice for query synthesis.
Effect of diversity regularization. We evaluate the effectiveness of the proposed diversity reg- ularization term in (6). with results presented in Fig. 4b) Incorporating the entropy-hased term, PaPQS achieves 10% improvement in RMSE over its non-diversity counterpart during the first two iterations, while maintaining comparable performance in the latter two stages. This reflects the utility of the diversity term scenarios with limited computational resources, where only small number of labeled PDE simulations can be acquired and each query must be highly informative. As shown in the bottom panel of Fig. 4bb] removing the diversity term causes the acquisition function to concentrate samples near the boundaries of the viscosity space regions that typically exhibit high epistemic uncertainty but yield limited incremental information. This sampling bias results in poor coverage of the parameter space and hampers surrogate model training. In contrast, the proposed entropy-based regularization encourages broader and more balanced exploration across the viscosity domain, facilitating the acquisition of diverse dynamics and reducing redundancy in the training data. This leads to more sample-efficient learning and faster generalization with fewer labeled simulations.
Effect of update steps. We conduct a parameter sensitivity analysis to investigate how varying the number of gradient update steps Natep affects the performance of our PaPQS framework. As shown in Fig. 4c increasing the number of update steps from 10 to 50 yields minor improvements in RMSE on both the Burgers' and KS equations. This indicates that small number of steps (e.g.. Nstep 10) already achieves strong performance. Notably, the time complexity of the gradient operator scales linearly with the number of update steps. practice, this parameter's setting can be flexibly adjusted based computational resources and target performance, enabling practical trade-off between efficiency and efficacy.
Conclusion
present PaPQS, a plug-and-play active learning framework that synthesizes informative PDE settings through flexible exploration in continuous design space, different from existing pool-based AL methods for neural PDE solvers. By jointly optimizing expected information gain and batch diversity, PaPQS enables uncertainty- aware, adaptive query synthesis instead of relying fixed candidate

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pools, accelerating the training of neural PDE solvers. Experiments across diverse PDE systems and surrogate architectures show that PaPQS consistently improves data efficiency, generalizes across acquisition strategies, and achieves consistent gains in performance and runtime. While PaPQS performs well on simpler PDEs (e.g., Burgers' equation), its benefit more modest on tightly coupled multi-field systems (e.g., 2D compressible Navier-Stokes). Future work may explore refined update strategies (e.g., adaptive step sizes or physics- informeed regularization) to further improve such complex dynamics and extend active learning strategies to the temporal domain.
6
Acknowledgment
Z.W. and X.N.Q. acknowledge the support from United States National Science Foundation (NSF) grants AMREF-2119105 SHF-2215573, and IIS-2212419. J.G., B.J.Y,, N.M.U., and X.N.Q. acknowl- edge the support by the U.S. Department of Energy's Office of Science Biological and Environmental Research (BER) program under project B&R# KP 161601017 and FWP#CC140, and Advanced Sci- entifie Computing Research (ASCR) under projects B&R# KJ0401010/FWPCCC 30 and B&R# KJ0403010 and FWP#CC138.
Many of the numerical experiments were conducted using advanced computing resources provided by Texas A&M High Performance Research Computing.
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Answer: [Yes]
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We explicitly discuss the limitations of PaPQS and directions for future work
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We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if' applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: This research does not involve the use of large language models as an original, important, non-standard component.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to our LLM policy (https://neuri)) /att::///earrrpsscc//oooerrreeee /c/////oreeeeeeee/222////.. for what should or should not be described.
19

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A Theoretical Analysis
A.1 Claim
Suppose predictive variance is used as the acquisition function. Then, preferred regions may reffect high prediction uncertainty, suffering from strong chaos without necessarily providing information about 0. In contrast, EIG explicitly quantifies the informativeness of to with respect to updating the model parameters, thereby better supporting robust and data- aata-efficient neural PDE surrogate learning.
Theoretical insight on linear case. To provide analytical intuition, consider simplified linear surrogate model
U tbw + e,
where & denotes the design variable (e.g., initial condition), ((, Ew.) represents the model parameters, and ~ (0, o²) is the additive noise capturing aleatoric or measurement uncertainty.
The expected information gain is defined as the mutual information between and w given ab:
EIG bb) I(w;
bb) Hu 4/) H(u w, w.zb).
Under the linear model, we have
we ab ((bu,, o²), w ~ N1 Ew), which implies the marginal distribution Hence, the conditional entropies can be computed as
u +b (0, a + o²).
(uu w. +b) log (2TeO))) H(u v)) log (2 + oo))). Subtracting the two gives
EIG( = H(u v)) H (u
w. ab)
log 2T 2b
Ew
+ +0))
log (
16 1b + +
2 log
Ø²
log 1
Dne tb o
This closed form shows that EIG( (b) depends purely on the epistemic uncertainty term E. T while the additive noise variance ²² only acts as a normalization constant in the logarithmic term. Similarly, the predictive variance can be expressed
Vlu Φ/} Vn, [bbw]
V. = E. 2/
+ 2
Unlike EIG({b), the predictive variance includes both epistemic and aleatoric contributions, and can therefore be inflated by stochastic noise unrelated to model learnability. This distinction clarifies why principled acquisition objective for active learning- it explicitly measures the
EIG serves expected reduction in parameter uncertainty. In contrast, predictive variance may overemphasize chaotic or noisy regions.
As for the optimization property, since En, > 0 ensures that the quadratic form e.. is convex and the outer function log(1+ æ) is strictly concave and monotonic, the composition EIG(b) defines concave objective. Thus, maximizing EIG(bb) corresponds to convex optimization problem (maximization of a concave function), for which gradient ascent guarantees convergence to the global optimum in this linear regime.
Although the exact closed-form EIG intractable in nonlinear PDE surrogates, this linear analysis provides theoretical support for the gradient-based optimization strategy adopted in PaPQS. It also explains why the ElG-based critic naturally filters out uninformative, noise-dominated regions and promotes robust and data-cfficient query synthesis in complex PDE settings.

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Illustrative example. To support our claim that the expected information gain (EIG) is more re- liable than predictive variance in selecting informative samples under chaotic PDE dynamics, we present both visual and quantitative evidence based on active learning case study of the Ku- ramoto- Sivooo--ivss (KS) equation. The detailed settings are as follows:
PDE: consider the ID KS equation. a canonical model exhibiting spatiotemppral chaotic 8y äe + 0x + v
drr The equation is solved under periodic boundary conditions on a spatial domain of length L = 64, discretized using N₂ = 256 spatial grid points. Time integration performed over N, 800 steps with fixed time step size A/ = 0.01;
öt dynamics, defined as du + u @u
Initial Condition (IC) ub: the viscosity parameter v is sampled from the range [0.05, 0.1]. sum of random sinusoidal modes induce variability and richness in dynamical behavior, which could be formulated as u = Σi sisin sin (2rk,zz L. + Φ;). Here, No. is the wavenumber, A; is the amplitude ranging from [-1, 1]. kę is the frequency sampled from [1, 10). and di is the phase from [0, 27.. We perform uniform sampling for all parameters to determine initial conditions;
u constructed
Surrogate model Mg: We adopt a probabilistic autoregressive neural network as the surro- gate model learn the evolution dynamics of the KS equation. Given the state u, time step I, the model predicts the distribution of the next state u++1 by outputting the mean and log- log-variance of a Gaussian distribution, i.e., (u (u,) = =N()o diag(oğ) (u)))), where both (.)()) and log (-) are parameterized by shared multilayer perceptron (MLP) with two hidden layers and dropout regularization. The model is trained with a Gaussian negative oigliikelhood (NLL) loss:
N₂ N.
(u(t + 1.8) plt 1, 0² 1,
DNLL =
log 2 (t+ +
(14)
To reduce computational cost while preserving temporal dynamics, we train the model to predict an up-sampled trajectory by selecting every 40 frames from the original simulation, resulting in 20 autoregressive prediction steps.
To directly compare the difference between two acquisition functions, consider pool-based active learning scheme, which follows the steps below:
1. We randomly sample 30 trajectories from the PDE simulator to form initial labeled set. 10 trajectories are used to train the surrogate model, and another 20 trajectories are held as the test set;
2. Next, construct candidate pool of 200 unlabeled samples by varying the ICs within the prescribed design space;
3. Each candidate is scored using one of the two acquisition functions predictive variance or EIG - and the top 10 samples are selected and added to the training set. Their estimation methods are provided in A Algorithms AT and A2
4. The surrogate model is then fine-tuned on the augmented dataset, and its performance is evaluated on the test set.
This single-round selection process enables a direct and fair comparison of the informativeness of the two acquisition strategies under identical conditions.
We repeat this experiment 10 times with different random seeds and report the RMSE and 95% confidence interval after training on the selected PDE data. The variance- method yields an RMSE of 4 10116.. while the EIG-based selection achieves a lower RMSE of 4.688=0.092 The EIG-guided AL achieves lower errors. indicating more effective sample selection for model improvement. We further visualize ground-truth PDE trajectories selected by predictive variance and ElG-based acquisition functions in Fig. Ao investigate the types of dynamics favored by each method. We observe that predictive variance tends to select highly chaotic trajectories characterized by rapidly changing modes, spatial discontinuities, and temporal irregularities (e.g.. samples 19, 141, and 42). This is expected, as predictive variance reacts strongly to output variability, which is amplified in chaotic regions. However, such samples despite their high uncertainty may be

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Algorithm A1 Predictive Variance Estimation via Dropout Sampling
Input: Initial condition tb, surrogate model Ma with dropout, number of samples N., prediction
horizon N₂
Output: Predictive variance estimate o
for E [3] do
2: Randomly sample a model Myg) by Dropout
3: Get mean and variance predictions {ul (t, (+)) (0) (t, x) he(1,))) via autoregressive rollout
4: end for
5: Get epistemic variance by computing the empirical variance of mean predictions, i.e., o opistemic
Var, ([u())))
6: Get aleatoric variance by averaging the variance prediction, i.e., o L
"aleatoric + E, (x) L.NN
7: Get total variance by adding the epistemic and aleatoric variance: o total f- epistemic alcatoric
2
8: return iotal
Algorithm A2 Expected Information Gain Estimation via Nested Monte Carlo
Input: Initial condition 4b, surrogate model Ma with dropout, number of outer samples N,, number of inner samples Nm prediction horizon N;
Output: Estimated expected information gain EIG($)
1: Initialize total EIG estimate EIG +-
2: for E [1, N.]
3: Sample model MM by Dropout
(1:0)² 4: Get mean and variance predictions (t, o (t,))) (t,x)HEE).N) via autoregressive roll-
out
5: Sample trajectory (+)) from (s (,,)))
6: Compute log-likelihood via log p(
(u (+) (0,00) + log (uls) (l000), o
7:
for E [1, N.. do
8:
Sample another model M! by Dropout
9: Get mean and variance predictions (f( (t, (am)) (t, (t, hEE(UNN via autoregressive
rollout
10: Compute log-likelihood via log p (()(0))0000m + log log((((();; (s(..)) (s.m)))
11: end for
12: Estimate marginal log- ligeeiill by log (((())) Em p(u()) 1a.m)
13: end for
14: Estimate the EIG by EIG(1) + 1 N, En', log (()))
return EIG(W)
less informative for model training due to their intrinsic unpredictability. In contrast, EIG tends to prioritize relatively stable trajectories that are still dynamically rich but exhibit more learnable patterns, avoiding excessively chaotic regions that may hinder learning. This allows the surrogate model extract more meaningful gradients, leading to faster convergence under limited sample budget.
A.2 Proposition
Let neural PDE surrogate model parameters be partitioned as 0 (0foat Oont), where Oun is fixed and only 0m is sampled. Assume that the dominant sources of epistemic uncertainty of the model can be captured by Oan [47]. Then, valid lower bound on the EIG can be estimated by sampling only A 0mn given by: EIG +b 2 Ep( [T (0 u)] - e (p((0))))))
Te TT(00m))))
Proof. We start from the definition of expected information gain (EIG) under the full model parameter space 0 = (Oreat, 0. Given input condition ab and model prediction u, EIG is defined as:
( Ee-pt()
[KL (p(u) 0, (b) || p(u +))]

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(a) Predictive variance selection
(b) EIG selection
Figure Al: The visualization of ground-truth PDE trajectories selected by different acquisition
functions.
where p(u v) = p(0)p(u 0, bb) the marginal predictive distribution.
Now, we assume that the feature extractor parameters fea are fixed to some constant 6 This implies that the prior over parameters becomes:
p(Oreat. Dout) = 8(0fcat Oraa) p(oout)
(15)
and the predictive distribution simplifies accordingly:
p(u 0, xb) = p(u 0 Oons 3b) = p(u Oot, 2b).
(16)

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Substituting this into the definition of EIG, we obtain:
(b) = [O (m (mm)p [L (p(u Oreat, Oout, 2b) ll p(u [b)]]
/ ʃ 6(Oreat - (eem) p(foui) KL (p(u
Breat: Bout +) || p(u 2b)) od dd
f p(oou)) KL (p(u Ofaar Our 2b) p(u <b)) out f
p((mm) KL (p(u Oonts vb) p(u xb)) d00
(p [L (p(u O bb) p(u eb))]
Mutual- Inf Inf inforrmaiion(( out, u 2/)
Epp(....)) u) (1o(oout u)] (pp(mm)p())) To (o(oon.))).
(17)
This completes the proof.
B Algorithm of PaPQS
The complete algorithm the proposed PaPQS is provided in Algorithm B1
Algorithm B1 The proposed PaPQS active learning framework.
1: Randomly sample Nisitiol IC batches and simulate their PDE solutions to build the cold-start
training dataset Suain.
2: Train neural surrogate Ma with Susin by SWAG to update surrogate parameter distribution p(0)
3: for itr active learning iteration do
4: Obtain 2 x Ninitial batches ICs (())
1 Nusa J. 5: Obtain the samples of f by Eq. (10) and joint samples {(O,, u(3b,, t; t;),) 0; ~ p(0), vb, e
ST00 E [1 N.)).. Permute joint samples get independent samples.
6:
Train neural critic function (-).) by maximizing Eq. (9)
7:
for each Seurch do
8:
for n E total update steps Nateps do for each E Ne] do Predict u (b (hthha l; 0) and estimate EIG by Eq. (9).
9:
10:
if ( (1, [N, Ne) then
12:
Calculate the gradient by Eqs.
end if end for
14:
Average the gradients across time steps.
16:
Conduct the gradient ascent by Eq.
Calculate the policy score by Eq. and determine whether to retain each update.
18:
end for
19:
end for
20:
Simulate all updated batches of ICs to get the ground truth and add them to Smain
21:
Train neural surrogate Mg with Swain by SWAG to update parameter distribution p(0)
22:
end for
C Details of PDEs and Neural Surrogates
In this section, we describe the neural PDE solvers evaluated in our study in Sec. 4 of the main text, along with the procedures used generate their initial conditions and corresponding hyperparameters. We note that our settings are aligned with the AL4PDE benchmank [3] for consistency and fair comparison.
hhunss//tthub.ccm/dduussaaaaaa/u
5

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C.1 PDEs and their parameters
Burgers' equation. We consider the dimeensioonll (ID) viscous Burgers' equation, canonical nonlinear PDE used to model wave propagation and shock formation:
0,u + u a. u U
(18)
where u denotes the velocity field, and v > is the viscosity coefficient. The equation equipped with periodic boundary conditions and initialized by prescribed function u(x, 0). Following previous benchmarks [13 23 28]. we sample PDE parameters on a logarithmic scale. Specifically, each parameter A is first uniformly sampled from [0, 1) and then transformed to its domain [a, b) via
ag exp
log
11
(19)
Kuramoto-Sivashinsky (KS) equation. The KS equation is nonlinear fourth-order PDE that models spatiotemporal chaos in systems such as flame fronts and thin film flows. one dimension, takes the form:
a,u + ud agzu + 1/ Uzeee u = 0,
(20)
where u denotes the scalar field over space and time, and v > controls the strength of the fourth-order dissipative term. The KS equation exhibits complex nonlinear dynamics and serves as a canonical testbed for studying spatiotemporal chaos and instability-driven pattern formation. Following the AL4PDE [3] configuration, the dissipation coefficient is treated as varying parameter, sampled uniformly from the range v [0.5,4] 4).
Combined equation (CE). The CE is nonlinear conservation law that generalizes several classical PDEs by combining nonlinear advection, diffusion, and dispersion 43) It is formulated by:
0,u 02 (ou² 3 B. y 0
(21)
where u denotes the scalar field, and a, 3, and are scalar coefficients. By choosing different tuples (ox, 7), this formulation recovers well-known equations such as the Heat equation with (0, 1, 0), Burgers' equation with (0.5, 1),), and Korteweg-de- Vriess equation with (3, 0, 1). We follow the standard setup adopted in AL4PDE [I3] that (e,18,7) are sampled uniformly from their respective ranges, with E (), 3 [0, 0.4), and 7 [0. 1).
Compressible Navier-Stokes (CNS) equation. The CNS equations govern the evolution of compressible viscous fluid by conserving mass, momentum, and energy [28] It can be written in the non-conservative form as:
0. (pv) 0,
(22)
p + Vv) -p+nnv (s+ !! 3/
V(V V(V.v),
(23)
a (c +
+ V. [p+ct 20l1v[12) v - v 0.
(24)
Here, the equation has four fields for 2D system, P denotes the density, v is the velocity vector including two components, and is the pressure. The viscosity terms involve shear viscosity n) and bulk viscosity Ç, and o' denotes the viscous stress tensor. We follow AL4PDE [13] and sample PDE parameters and ( independently from log-uniform distribution in the range [10 10-1).
C.2 Initial conditions
Initial conditions are generated using the force density method (FDM) based JAX simulator and the IC generator from PDEBench 28 Each initial condition is constructed as superposition of sinusoidal waves:
(x)
Σ
No. A, sin
2TTkE L
pp: )
(25)
where N., denotes the number of wave components, I is the spatial domain length, and A,, k; and (p; represent the amplitude, wave number, and phase of the i-th component, respectively.

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For the Burgers' equation, we set No =2 The amplitudes A; and phases yi are sampled uniformly from [0. 1) and [0, 2x). respectively, while the wave numbers k; are integers sampled from [1,5). In addition, we apply a windowing operation by setting all values outside the interval [2r,, .: to 7ero with a probability of 10%. Here, zr, is sampled uniformly from (0.1, 0.45) and on from (0.55, 0.9]. Furthermore, the sign of u is flipped randomly with a probability of 10%.
For the KS equation, we set Nw 10. The amplitudes A, and phases yp; are sampled uniformly from [) and [0, 2T), respectively, while the wave numbers k, are integers sampled from [1, 10).
For the CE, set N 5. The amplitudes A; and phases yi sampled uniformly from [-0.4, 0.4)
and [0, 2m), respectively, while the wave numbers k; are integers sampled from [1,3).
For the 2D-CNS equation, the initial condition of each physical field (velocity, density, and pressure) constructed as superposition of 2D sinusoidal modes:
u (x, pO (I. 8). (z)
Σ
Ake sin (2T(koT + kyy) + (ke..ty)
(26)
where = (- ked x 1- Rioss Asd \ ((kks (iy) K2 = 0 or ki = 0}. We set Kio = 4. For each mode, the amplitude is set deterministically as Abe = 1/ / 1-2 + kg. The corresponding phases Pk are uniformly sampled in the range [0, 2T). After summing over all modes, the velocity magnitude is normalized to match a sampled target Mach number from [0.1, 1). Then, we enforce the positivity the density and pressure channels. The raw density field px(x,, y) is rescaled to a positive field as
(xx v) = P 1 4p (( v)
(27)
maxz,y (o, v)
where p is the base density sampled from (0.110)) and A, from (0.013, 0.26). The pressure field (r, 2) is transformed analogously using its own scale Ap sampled from [0.04, 0.8), and offset f Tp, where T is sampled from [0.1, 10). Finally, we apply a windowing operation by setting all values outside the interval [0, 1] to zero with a probability of 50%.
C.3
Grids
We consider neural PDE solvers that require discretizing both the spatial and temporal domains. The domain length and grid resolution configurations for each PDE are summarized in Table CI
Table C1: Domain lengths and resolution configurations for each PDE. Simulation Res. denotes the resolution settings used in the numerical simulations, while Neural Surrogate Res. refers to the resolution settings used during neural PDE surrogate training.
Simulation Res.
Neural Surrogate Res.
PDE
Temporal Domain
Spatial Domain
N₂
(Ne, (Ny)
N
(Nz, (Nyl)
Burgers
[0, 2]
[0. 1]
201
(1024)
41
(256)
KS
[0, 40]
[0, [0.1,100)]
801
(512)
41
(256)
CE
[0, 4]
[0, 16]
501
(64)
51
(64)
2D-CNS
[0, 1]
[0, 11 10, 11
(128, 128)
21
(64, 64)
C.4
Neural surrogates
U-Net. We employ an enhanced U-Net architecture [5] as one of the neural PDE surrogates. This architecture improves the traditional U-Net architecture 48] by incorporating recently developed components with improved performances in computer vision tasks, including Wide ResNet-style convolutional blocks [4), group normalization [50], and spatial attention mechanisms [5T]. additionally replaces max- moollioning with downsampling layers to improve multi-scale feature extraction. To further enhance its capacity for modeling complex physical dynamics, it introduces a Fourier U- Net variant, which integrates Fourier Neural Operator layers [44] into the deeper encoder and decoder blocks. This hybrid design allows the model to efficiently capture both global and local spatial dependencies via fast Fourier transforms and mode-specific weight multiplication. Compared to
7

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standard U-Nets, this architecture leverages Fourier representations to effectively capture multi-scale PDE dynamics, enabling generalized modeling across diverse physical systems.
Fourier Neural Operator (FNO). We employ FNO [44] as a neural PDE surrogate due to its efficiency and widespread adoption in scientific machine learning. FNO is designed to learn mappings between infinite-dimense dimensioaaa function spaces, enabling efficient solutions as surrogate modeling of complex PDEs. Unlike conventional neural networks, FNO operates in the Fourier space by parameterizing the integral kernel and applying transformations via the Fast Fourier Transform (FFT), allowing efficient modeling of global spatial dependencies with quasi-linear complexity. Its architecture lifts input functions to high-dimensional spaces, processes them through Fourier layers and nonlinear activations, and projects them back to the output domain. FNO supports mesh invariance, resolution generalization, and efficient inference, making it well-suited for modeling complex PDEs across a wide range of scientific applications.
SineNet. We employ SineNet 7 as another neural PDE surrogate. Conventional U-Nets, while effective for multi-scale spatial processing, suffer from temporal misalignment in skip connections for latent feature evolution across time. SineNet mitigates this issue by stacking multiple U-shaped network blocks called waves each advancing the solution over small temporal interval. This multi-stage design reduces misalignment and improves modeling accuracy. Each wave combines sequential and parallel scali processing through disentangled block residuals, enhancing expressiveness while maintaining parameter efficiency via adaptive channel widths. Its ability to capture complex temporal dynamics and support variable time steps makes SineNet a robust choice for scientific modeling.
Hyperparameters. We follow the settings provided in the AL4PDE benchmark 13 For U-Net, use the GELU activation function [52] and Fourier-based conditioning [51]. with a channel multiplier of [1, 2, 4] and 16 hidden channels, resulting in 3.38M and 9.18M parameters for ID and 2D versions, respectively. The implementation of FNO also adopts the GELU activation but uses an additional input channel for conditioning. It consists of 4 layers, with a width of 64 in ID and 32 in 2D cases, respectively. The number of Fourier modes is set to 20. The parameter counts are 0.68M (ID) and 6.56M (2D). Lastly, for SineNet, we apply the GELU activation and Fourier conditioning. with 32 hidden channels and four sequential U-Net "waves". The 2D version of SineNet contains approximately 5.02M parameters. Par
D Further implementation details of the proposed PaPQS
D.1 Initial condition search space
To enable gradient-based optimization, we restrict our search space to be continuous and differentiable. The search space includes all PDE parameters and continuous scalar parameters used in constructing the initial states. Discrete or non-differentaabe parameters (e.g.. wave numbers and spatial windowing bounds) are excluded from the optimization process. For the Burgers' equation, we include viscosity v. amplitudes A, and phase yp. For the KS equation, we include viscosity v, domain length I., amplitudes A. and phase y. For the CE, we include coefficients a, 3, and 7, amplitudes A, and phase sp For the 2D-CNS equation, we include shear viscosity 17. bulk viscosity S, phase yp, Mach number, base density f and its scale Ap, as well as base pressure and its scale Ap.
D.2 Neural critic function
We design a neural critic function (9, ult, v)) based on the Mutual Information Neural Estimation (MINE) framework (39 to estimate mutual information (equivalent to EIG as discussed in the main text) between the last-layer parameters of neural surrogates and trajectory predictions. The critic function learns to assign a score paired samples (0, u(t, vf) by approximating their log-density ratio. It consists of two input branches; parameter encoder for 0 de and trajectory encoder for utt, .)) E IR XxNN where X' is 1D or 2D spatial grid coordinates in our study and N is the number of physical fields.
The O-branch a multilayer perceptron (MLP) composed of two linear layers with the ReLU activation functions that projects the input into a compact latent representation. The u-branch employs either a ID or 2D convolutional encoder depending on the dimensionality of the input. They both apply three successive convolutional layers followed by the ReLU activations to extract
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hierarchical features from frames of PDE trajectories. The resulting features are flattened and passed through a dense projection head match the dimensionality the f-branch output. The outputs of both branches are concatenated and passed through fusion MI.P that outputs a scalar critic score. The detailed layer information and hyperparameters are summarized Table[D]
The entire critic neural network is trained by minimizing the following objective:
(p (T (6, u)] - c Ep(0) (o
(28)
We train this neural critic function using mini- batch stochastic optimization on set of joint samples (0, u). The full sample set is randomly split into 80% training and 20% validation subsets. We use the Adam optimizer [33 with standard settings to minimize the MINE loss. At each active learning iteration, the neural critic function is trained for maximum of 500 epochs. To prevent from overfitting and reduce computational cost, we employ early stopping, terminating training if the validation loss fails to improve for 20 consecutive epochs.
Table DI: Overview and hyperparameter configuration of the neural critic function. We set d₁ 128 our study and refers to the kernel size of the convolutional layer.
Component
Layer Type
#Hlidden Channels
Output Dim
Linear ReLU
da dn d, dn//2
da da/2
0 Branch
Linear ReLU
Conv ID ReLU
ks=3, stride=2, No - dr/61 ks=3, stride=2, dn/ /n//61 dx/32
(d,/61, de d2/2)
Branch (ID)
Conv ID ReLU
(d,/32, dz/1)
Conv ReLU
ks=3, stride=2, dn//32 //22 dn/10
(dx/16, ds
Conv2D ReLU
ks=(3,3), stride=2, No No dn./32
(dx,/32, d, 12, d,,
u Branch (2D)
Conv2D ReLU
ks=(3,3), stride=2, dx/32 + da da/32 ks=(3,3), stride=2, dn /32 + d
(ds/32, de 1,.4,,0 dz/4) (dn/16, de 18, dy/8)
Conv2l) ReLUU
Linear ReLU
flatten size + flatten size/4 /4 flatten size/ /4 + di.
flatten size /4 di dn/2
Flatten Layer
Linear ReLU
Linear ReLU
d, d2//2
Linear ReLU
dn dn dn dn/4 dn/4
dn
Fusion MLP
Linear ReLU
di. /4
Linear
D.3
Hardware and Platform
All experiments are conducted on single GPU node of a high-performance computing cluster. The node is equipped with an Intel Xeon 6248R CPU and an NVIDIA A100 GPU with 40 GB of VRAM. Simulating, training. and evaluation are performed using Jax and Py PyTorch frameworks on Lisex-based operating system.
E
Additional empirical results
E.1
Detailed results in Fig. of the main text
we provide all the RMSE and standard deviation values different quantiles by neural PDE surTogates for different PDE systems, which have been used to prepare for Fig. 2i the main text. For the Burgers' equation, each method was evaluated over 10 independent runs with different random seeds. while for the other systems, results are obtained over 5 runs. Again, it is clear that applying PaPQS can consistently achieve the best or second best approximation to the ground-truth PDE solutions across nearly all tested PDE systems.
In Tables ET
E.2 Exemplar neural PDE solution approximations
We present several representative case studies illustrate the effectiveness of our PaPQS active learning framework. Specifically, compare prediction results with and without PaPQS on two

---

PDE systems: the KS and the 2D-CNS equations. These systems are selected due to their complex dynamics, chaotic behavior, and high nonlinearities. which pose significant challenges for neural surrogate modeling. The visual comparisons in Figs.F]] F4 F highlight how PaPQS improves prediction accuracy and stability these scenarios through gradient-guidee query synthesis iterations.
Table El: Error metrics Burgers' equation.
Iteration
RMSE x10 -
Random
3.684.1..203
3.278:12.107
1.00710:485
1.062.10.614
0.552.10.133
SBAL
3.684:11.203
1.179:1.2233
0.58620.116
0,40010.075
0.259:10.028
LCMD
3.684:11.200 0.808=00.053
0.521
0.39410.043
0.269:00.01
Core-Set
3.684:1..203
1.061100160
0.65910.100
0,476.1.134
0.292-10.015
Top-K
3.684:11.203
1.494
0.964
0,477=0.044
0.360:1000%
BAIT
3.684+1.2000
0.903+001138
0.537=0.030
0.392=0.035
0.266=0.0024
LHS
3.441
1.930+00.00
1.354
1.057=0.539
0.521
Random+PaPQS
3.684+1.2200 1.316=0.712
0.80
0,531
0.358=t0.115
SBAL+PaPOS
3.684+1.200
0.902+0.342
0.515=00088
0.334+0.042
0.231+00022
LCMD+PPPOS
3.684+1.203
0.757+0.226
0.109
0.379+0.101
0.235++00018
50% Quantile
x10-²
Random
0.182:10.015
0.122:00.015
0.083=0.010
0.05810.005
0.044
SBAL
0.182::0015
0.178:00.032
0.105=0.011
0.07810.011
0.054
LCMD
0.182::0.015
0. 0.129+0.014
0.101
0.068
0.050+0.006
Core-Set
0.182+:00015
0.169+0.017
0.133+0.013
0.094+0.014
0.063++0.008
Top-K
0.182:10.015
0.197-10.020
0.176
0.109
0.078:10.012
BAIT
0.182-10.015
0.15011.014
0.115100011
0.07910.006
0.058110.008
LHS
0.174:10.14
0.116:10.014
0.081=0.009
0.062=0.007
0.054=10.011
Random+PaPQS
0.182::0.015
0.119+0.010
0.072=0.007
0.054=0.008
0.043
SBAL+PaPOS
0.182+0.015
0.163++0.031
0.099+0.017
0.078+00008
0.059++0.008
LCMD+PaPOS
0.182+00005
0. 0.129-10.038
0.106:10.019
0.094
0.052-10.007
95% Quantile
x10-²
Random
1.468:0.116
0.83440.125
0.502=10003
0.34310.014
0.255+0.025
SBAL
1.468+0.136
1.054=0.224
0.544=0.065
0.409#0.064
0.269+0.026
L.CDD
1.468++0.133 136
0.669+0.009
0.526+00004
0.347
0.259+0.032
Core-Set
1.468++0.134
0.865-10.123
0.662
0.503
0.259-10.020
Top-K
1.46810.113
1.273-10.177
1.04510.200
0.57510.064
0.449.10.077
BAIT
1.46:10.136
0.800-10.160
0.532=10.05
0.378
0.274-10.026
LHS
1.390
0.803+0.114
0.474=0.038
0.344=0.027
0.246:10.024
Random+PaPOS
1.468-10.136
0.762-10.075
0.473-10.883
0.024
0.265110.039
SBAL+PaPOS
1.468+0.136
0.889=10.178
0.503=0.095
0.396=0.073
0.277+10.045
LCMD+PaPQS
1.468+0.136
0.683+:0.189
0.495=0.111
0.456#0.100
0.261+0.022
99% Quantile x x10-2
Random
6.315+:0838
3.327=t.724
1.653=0.111
0.968+0.046
0.649+0.027
SBAL
6.315++0.388
3.169+0.945
1.360+0.221
0.987+0.239
0.599-0.0056
LCMD
6.315-+0.838
1.802++0.157
1.2237
0.819
0.573++0.041
Core-Set
6.315110.838
2.461-10.500
1.756110.300
1.153
0.703-
Top-K
6.315:10.838
4.456111.685
3.259=11039
1.347+0.112
L.048110.266
BAIT
6.315::0.338
2.371=:0.118
1.255=0.100 108
0.853+0.066
0.612+:0.055
LHS
6.215+1.012
3.017=20.476
1.515= 109
0.963+0.046
0.650+00.077
Random+PaPQS
6.315::0.338
2.756=111441
1.573=0.387
0.943 146
0.631
SBAL+PaPOS
6.315+0.838
2.182+00.74
1.177=0.232
0.867+0.262
0.606:00.05
LCMD+PaPQS
6.315++0.88
1.779-L0.376
1.114-0.164
1.02440.241
0.571-40.038
10

---

Table E2: Error metrics on KS
Iteration
2
3
RMSE
Random
0.452
0.370:10.012
0.312:10.013
0.272:10.010
0.229:10.010
SBAL
0.452=0.026
0.347=0.020
0.2811000100
0.236:10.008
0.200:10.012
LCMD
0.452.10.026
0.370.10.009
0.315100.03
0.266
0.219110.018
Core-Set
0.452=0.028
0.389-00.011
0.335+00013
0.008
0.235-10.020
Top-K
0.452=1.022
0.378
0.3051000011
0.264
0.225:10.015
BAIT
0.452 026
0.3681100117
0,29410.011
0.240:L0.009
0.205
IHHS
0.439= 0.008
0.369+0..224
0.316+0.011
0.270+
0.222++0.012
Random+PaPQS
0.452=0.022
0.344+00019
0.298+0.012
0.246+00.009
0.203+000007
SBAL+PaPQS
0.452=0.028
0.313:0.017
0.261
0.221
0.183:10.009
LCMD++aPPQ
0.452=0.026
0.354+0.011
0.315+00009
0.247+00019
0.201=00.010
50% Quantile
Random
0.021=0.005
0.001
0.00810.001
0.005
0.003:10.001
SBAL
0.021=0.005
0.016+0.004
0.013+00003
0.008+00.001
0.006++0.001
LCMD
0.021=0.005
0.020=0.003
0.016:00.003
0.009+00.003
0.006:10.001
Core-Set
0.021=0.005
0.020+0.003
0.016+00.003
0.009+00.003
0.009+0.002
Top-K
0.021=0.005
0.020#0.003
0.018+0.002
0.012+t0.003
0.010+00.002
BAIT
0.021=0.005
0.020#0.003
0.015:10.003
0.008=0.001
0.005:+0.001
LHS
0.019=0.001
0.011+0.002
0.007+:00001
0.005
0.003-+0.001
Random+PaPOS
0.02111.005
0.010.10.001
0.007110.001
0.005
0.000
SBAL+PaPOS
0.021=0.005
0.014+0.000
0.012+0.001
0.008+00.001
0.006++0.001
LCMD+PaPQS
0.021
0.017110002
0.01410.002
0.008
0.006.10.001
95% Quantile
Random
0.603=0.10 106
0.36310.020
0.231110.024
0.14310.011
0.094-10.006
SBAL
0.603
0.376.10000
0.25510.031
0.1191100118 0.163
LCMD
0.603 106
0.408-
0.34440.024
0.230-L0.035
0. 0.120--0.223
Core-Set
0.60310. 106
0.5015
0.425110.034
0.29510.022
0.213-10.053
Top-K
0.603
0.4581100117
0.34040.026
0.257
0.188-10.016
BAIT
0.603 106
0.45010.051
0.26910.043
0.163+10.20
0.100::0.012
LIIS
0.572:10.020
0.352.10.055
0.23811.027
0.0411
0.091-10.006
Random+PaPOS
0.603=0.1m 106
0.321
0.216:100030
0.039
0.085:00000
SBAL+PaPOS
0.603=0. 106
0.317+00031
0.221+00011
0.157+0.019
0. 105++0.009
LCMD+PaPQS
0.603 106
0.426=00050
0.331+:0033
0.198+0.0023
0. 0.132-0.017
99% Quantile
Random
2.368=0.152
1.844+0..00
1.382+00.17
1.040+0.092
0. 0.708+00.08
SBAL
2.368=0.155
1.655+0.133 137
1.077+001000
0.844 103
0.619-20.093
LCMD
0.358
1.811
1.440
1.151+0.123
0.802++0.149
Core-Set
2.368=0.153
1.920#0.077
1.571
1.230+10.046
0.982+0.202
Top-K
2.368=0.155
1.86010.122
1.356:0.092
1.13840.086
0.873+:0.119
BAIT
2.368=0.113
1.782+0.112
1.265++0.131
0.863+00051
0.607--0.055
LHS
2.296=0.053
1.844
1.426+0.009
1.036+00.02
0.667+0.058
Random+PaPOS
2.368=0.153
1.686+1. 173
1.33740.104
0.918+0.081
0.607-10.040
SBAL+PaPOS
2.368-0.113
1.422.1..150
1.046.10.061
1.778-0.000
0.546.10.066
LCMD+PaPOS
2.368-10.133
1.714
1.470+10.038
1.01140.144
0.7012

---

Table E3: Error metrics on CE.
Iteration
3
4
5
RMSE x10
Random
4.651=1.299
3.81411.111
2.609:10:466
1.630:10.257
1.108:10.117
SBAL
4.651=1.293
1.597-100083
0.931=0.125
0.496:10.087
0.318#0.048
LCMD
1.51111.993
L.22110.121
0.957.10.114
0.60910.117
0.0044
Core-Set
4.651=1.299
L.596+10223
1.033+00.06
0.761
0.424-
Top-K
4.651
L678.10.099
0.90411.101
0.529110.103
0.373:10.077
BAIT
4.651 1.299
L.415:0.187
0.900:10.102
0.660:10.159
0.424
LHS
5.130=0.808
3.626++
2.668+0.383
1.852+0.301
1.312+0.144
Random-PaPQS
4.651=1.299
3.332+1.170
2.113+0.278
1.334
0.916+0.008
SBAL+PaPQS
4.651=1.299
1.333:10.092
0.759:1.002
0.461
0.295=10.00
LCMD++aPOS
4.651=1.293
1.485+0.110
0.885+0.073
0.536+0.117
0.316+0.078
50% Quantile
x10
Random
0.238=0.025
0.16610.036
0.125
0.083
0.065=00.004
SBAL
0.23==0.025
0.200+00.024
0.125
0.076+00008
0.052++0.004
LCMD
0,238=0.022
0.171=00.007
0.128
0.083::00008
0.054=0.004
Core-Set
0.238=0.025
0.224+00.07
0.160+0.000
0.1099
0.083+000009
Top-K
0.238=0.0m 025
0.211
0.156
0.111=00015
0.073+0.006
BAIT
0,238=0.022
0.186:10.018
0.146:10.011
0.108:10.011
0.080=0.0066
LHS
0.249=0.030
0.145:10.022
0.117
0.085++0.011
0.066+000033
Random+PaPQS
0.238
0.173-10.020
0.118110:010
0.085.10004
0.063
SBAL+PaPQS
0.225
0.185++0.029
0.123++0.02
0.077+0.009
0.052+0.003
LCMD+PaPQS
0.249
0.186.10.013
0.131
0.079110005
0.058110.004
95% Quantile
x10
Random
2.373=0.220
L619910.22
1.09010.050
0.695::00399
0.516:0.0119
SBAL
2.323
1.723:10.166
0.980-10.070
0.51011.036
0.31310.014
LCMD
2.373=00.200
1.485
1.038+0.087
0.609+00001
0.301
Core-Set
2.37310.220
1.902-0.379
1.38910.1
1.102110.499
0.59810.009
Top-K
2.22
1.586.10.101
1.236:10.099
0.73910.111
0.489410.048
BAIT 2.373=0.220
1.56710.152
1.03810.085
0..000
0.405=0.005
LHS
2.537-1.2133
1.516.10.098
1.080
0.70910.057
0.530-10.013
Random+PaPQS
2.373=0.220
L666:10.214
1.016
0.717+0.112
0.505+00013
SBAL+PaPQS
2.373=0.220
1.588+0.118
0.919+0.099
0.515+0.048
0.303+
LCMD+PaPOS
2.373=0.220
1.495:0..110
0.992+0.073
0.566+0.015
0.371
99% Quantile x x10-2
Random
10.192++.5523
1.260
4.741++0281
2.893:+.2227
1.870+00.09
SBAL
10.192+1.522 1.523
4.756:10.215
2.701+:0251
1.333+001100
0.896+0.053
LCMD
10.192+
4.198
2.787++0.21
1.57140.212
1.036+0.006
Core-Set
10.192
5.056+10.821
3.526
2.638+1.066
1.446+0.292
Top-K
10.192+1.523
5,382-10.373
3.174410.81
1,75640.448
0.972+00092
BAIT
10.192+
4.290-10.307
2.896+
1.939+0.172
1.301 1041
LHS
10.785+1.740
6.863+0..77
4.778
3.09040.556
1.874+0.056
Random+PaPOS
10.192+1..22
6.959
4.30510.642
2.74740.187
1.741
SBAL+PaPOS
10.192.11.522 4.463-10.583
2.485.10.182
1.424.1.003
0.866
LCMD+PaPOS
10.59
4.364.10.204
2.727-10.13
1..18
1.025+0.022
12

---

Table E4: Error metrics on 2D CNS.
Iteration
4
5
RMSE
Random
2.632.10.339
2.162:10.029
1.856:10.100
1.572:10.072
1.362-10.065
SBAL
2.662:10.339
1.97911.222
1.790:10.203
1.488:0.144
1.205=00022
LCMD
2.669.10.339
1.991100293
1.734-10.188
1.356.10.081
1.277-10.083
Core-Set
2.332
2.322++0.350
1.731
1.613++0202
1.343
Top-K
2.662:10.339
2.16911..129
2.070-10.368
1.62310.524
1.313
BAIT 2.662.10.339
2.167:1..164
1.715:10.269
1.426:10.209
1.234 126
LHS
2.459=00.081
2.134++0.144
18899+0.098
1.514++0.059
1.344
Random-PaPOS
2.662:#0.339
2.104+0.144
1.780=+0.073
1.541
1.313=0.070
SBAL+PaPQS
2.662::0.339
1899:10.149
1.621=0.1511
1.408
1.174=0.041
L.CMD+PaPQS
2.662
1.943+0.258
16455+0.16
1.318++0.23
1.285
50% Quantile
Random
0.506=10.199
0.447=1.116
0.356+:0111
0.266=t0.087
0.209=0.033
SBAL
0.506+0.111
0.480++0.116
0.543+00.44
0.336++00063
0.295=000053
LCMD
0.506=0.119
0,574:10.36
0,412:+:.234
0.317:#0006
0.312=0.085
Core-Set
0.506+0.119
0.562++0.155
0,411++0.085
0.433++0.19
0.408
Top-K
0.506
0.653+0.166
0.521
0.483
0,400=0.065
BAIT
0.506=0.119
0.637:10.366
0.392:10.076
0.335=#0.069
0.311=0.093
LHS
0.553+0.132
0.503+10.068
0.304++0.035
0.264+00006
0.233+
Random+PaPOS
0.506:10.199
0.429.10.092
0.322-10.039
0.240
0.220-1.0061
SBAL+PaPQS
0.506+0..119
0.461++.111
0..014
0.265
0.259+0.080
LCMD+PaPQS
0.506.10.119
0.567-10.144
0.375.10.231
0.267.1.002
0.268
95% Quantile
Random
4.421
3.491110.154
2.828100.14
2.317=10.007
1.927=0.11
SBAL
4.421
3.308-10.550
2.936.10.370
2.310.10.349
1.821 10. 128
LCMD
4.621
3.263-100.61
2.758+00.351
2.025+:0.17
2.306
Core-Set
4.42110.630
4.23511.899
2.952-10.375
2.690-10.39
2.189-0.437
Top-K
4.421
5.009:12.442
3.891
2.911 1.392
2.238
BAIT 4.421
3.700110.263
2.783
2.238:104404
1.900=0.273
LHS 4.173.10.29
3.283-10.240
2.840.10.230
2.250.10.102
1.926-0.087
Random+PaPQS
4.421=0.630
3,479:10.15
2.797:40.281
2.267
1.932
SBAL+PaPOS
4.421
0.246
2.518+10.432
2.150++0:40
1.809
LCMD+PaPQS
4.421
3.361
2.640-10.33
1.961#0.225
1.905=0.118
99% Quantile
Random
11.378
9.135++0.253
7.754-+0.507
6.620++0.340
5.735=00320
SBAL
11.378=1 863
8.295+1.006
7.195:t0.786
6.058+0.573
4.933=0.112
LCMD
11.378 1.863
8.196-H0.926
7.229+00.009
5.569-40.322
5.269
Core-Set
11.378=1. 863
4.739+11.416
7.263++0.707
6.646440.74
5.404
Top-K
11.378=1. 1.863
11.42445.555 585
8.531 1478
6.466:42.101
5.237=0.417
BAIT
11.378 1.863
8.948-10.487
7.140++1.16
5.923++0.922
5.059+0.598
LHS
10.422=0.2m
8. 8.800 769
7.727=00.33
6.374
5.611=0.132
Random+PaPOS
11.378 1.863
8.85110..75 798
7.732+10.423
6.689-10.64
5.916=0.370
SBAL+PaPQS
11.378 1.863
8.275.10.722
6.911-10.750
5.985.10.879
1.857
LCMD+PaPOS
11.378 1.863
7.944
6.992-10.582
5.235-10.403
5.156
13

---

param: [1. (1.40363, 14.900932]
mse: 0.3510
Random
param: (1.40363, [4.90093]
mse: 0.0002
Random-PaPCa
param: [0.57211, 20.85272]
2.3304
ground Iruth
om
param: [0.57211, 20.85272]
mse: 1.1463
gound Iruth
preciction
enor
Random+ PaPQS
param:
[1.15515, 16.35942]
mse: 0.8308
groand
preciciion
Random
param: [1.15515, 16,35942]
mse: 0,4332
truth
Random+PaPoS
36
Figure E1: Neural PDE solution examples the KS equation. The left, center, and right columns display the ground truth, model prediction, and absolute error, respectively. The x-axis represents time, while the y-axis corresponds the spatial domain.
14

---

param: 50.58, 5.34662]
mse: 4.2238
prediction
Random
param:
(0.58082, 5.34662]
mse: 0.1753
grard buth
predition
Random+PaPOs
14-
42-
80
param: (0.62750,
78.31708]
mse: 0.9308
grund truth
predictoon
Random
82-
60
param:
[0.62 (0.62750,
78.31708]
mse: 0.2672
grurd
preiition
Random+PaPOS
param:
[0. (0.76629,
29.39786]
mse: 0.9039
geard trth
prediction
Random
param: (0.76629, 29.39786]
mse: 0.2151
grurd tuth
preiition
Fandom+PaPOS
Figure E2: Neural PDE solution examples on the KS equation. The left, center, and right columns display the ground truth, model prediction, and absolute error, respectively. The x-axis represents time, while the y-axis corresponds to the spatial domain.
15

---

param: [0.000197, 0.000814]
Random MSE: 51.159
Random= MSE: 40.460
Ground Trush
Flandom
Random- PaPOs
Random
Random PaPOs
[0.002719, 0.076279]
Random MSE: 0,0296
Random PaPQS MSE: 0,0125
PanQ
Ground
Random
01-3
1WB1O WV-0-11
Random-Parie PaPOS
Figure E3: Neural PDE solution examples on the 2D-CNS equation. Each column displays snapshots at time steps t 0,5, 5, 10, 15, 20. For each example, the first row shows the ground truth. The second and third rows compare neural surrogate approximations by the models trained with random sampling and PaPQS, respectively. The fourth and fifth rows compare the corresponding absolute approximation error.
16

---

[0.070046, 0,003489)
Random MSE: 3.8726
Random-PaPOS MSE: 1.0257
Ground
AAbrOt 9-1-8
Pandom
PaPOS
www. ww0.91 10--1 80-0
Ground Random-PPPPO
param: [0. 00.088633, 0.027022]
Random MSE: 0.8153
Random+PaPPO MSE: 0.4454
PROCA B1KE
BRO00 1910
K00 -115
01477
RandomuPaPOS
Figure F4: Neural PDE solution examples on the 2D-CNS equation. Each column displays snapshots time steps 0, 5, 10, 15, 20. For each example, the first row shows the ground truth. The second and third rows compare neural surrogate approximations by the models trained with random sampling and PaPQS, respectively. The fourth and fifth rows compare the corresponding absolute approximation error.
17


Reviews:
[{"review_id": "KEcbsdLyCh", "reviewer": "Reviewer_ZBUc", "summary": "The paper presents a plug and play approach towards improving neural network based models which act as PDE solvers. These models rely on training data which can be collected based on the approach in the paper PaPQS. It has two components which consider expected information gain (EIG) and sample diversity together to provide data to train neural PDE surrogates. PsPQS can be used in conjunction with other active learning strategy. Empirical results are presented to support the claims.", "questions": "1. Fig 2. is not possible to decipher, it is clumsy and the color scheme is bad. I am not able to decode which method is doing how much better than the other. Can you make it better and more readable?\n2. Where are the tables (numerical results)? It is mentioned about Table E1, E2? I don't see them in the main paper, the hyper-links don't work. Without numerical evidence, it is hard to justify the viability of the method.\n3. Fig 3(a) I don't see too much benefit over random.\n4. Fig 4(b) Importance of the diversity component is not prominent but it has been claimed otherwise. Can authors clarify and provide more evidence?\n5. Fig 4(a) The main EIG component also doesn't seem to be independent on it's own. Can you provide more evidence?\n6. Fig 2. Is there a reason to only use PaPQS with certain AL method and not others? Where is the curve for PaPQS alone?", "limitations": "Authors do not include any limitations of their work or any negative societal impact. I think the time complexity of the proposed approach should be discussed as it is expected to be a bottleneck when making an informed choice about the training data needed.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n1. The idea presented in the paper is very clean and makes a lot of sense intuitively, the authors do a good job in providing intuition for the idea.\n2. The experimental section answers a lot of interesting questions each of which have their own merit.\n3. Relevant literature has been covered.\n\nWeakness:\n1. Empirical evidence is weak to support that PaPQS has noticeable benefits.\n2. The presentation is hard to follow. I think a lot of things are mentioned in a non-standard manner. Please see questions section for more elaborate details.", "was_revised": true, "final_justification": "The Authors rebuttal answered my questions hence I have upgraded the score", "rebuttal": "We appreciate your thoughtful questions and feedback.\n\n> Weakness: Empirical evidence is weak to support that PaPQS has noticeable benefits.\n\nWe understand the reviewer’s concern regarding the empirical benefits of PaPQS. We respectfully note that the performance gains of PaPQS may appear visually modest in Figure 2 due to the varying error scales across the five active learning iterations. However, based on the numerical results reported in Tables E1–E4 in Appendix E, PaPQS consistently improves performance across all tasks and acquisition strategies. For example, Random+PaPQS achieves an average RMSE reduction of **48.68%** on the Burgers’ equation task, while SBAL+PaPQS and LCMD+PaPQS achieve average improvements of 15.60% and 8.18% on top of state-of-the-art active learning strategies, respectively. Across the remaining three tasks, the performance improvement is also consistent, with an average of 6.99% on KS, 12.27% on CE, and 3.79% on 2D-CNS.\n\nWe also emphasize that PaPQS is designed as a plug-in module that complements rather than replacing existing acquisition strategies. The fact that PaPQS consistently improves random sampling, uncertainty-based (e.g., SBAL), and diversity-based (e.g., LCMD) active learning methods across diverse PDE systems highlights its broad applicability. These results also demonstrate the feasibility of gradient-based search over continuous initial condition spaces in PDE problems, which can inform future active learning designs that more effectively exploit the structure of such continuous domains.\n\nLast but not least, to the best of our knowledge, PaPQS is the first query synthesis method for active learning of PDE neural surrogates considering continuous design space. To address the challenges in developing efficient solution algorithms, we estimate the acquisition function based on expected information gain (EIG) through the neural critic function with a 3-layer vanilla convolutional neural network and MINE loss, which is computationally scalable but consistently outperforms state-of-the-art active learning methods on our task. We believe that further enhancing the neural critic through more expressive architectures or refined training objectives could further boost the performance of PaPQS. We leave this as a promising direction for future work.\n\n> Question 1: Fig 2. is not possible to decipher, it is clumsy and the color scheme is bad. I am not able to decode which method is doing how much better than the other.\n\nWe understand the concern that the current figure may appear visually dense due to the number of methods and overlapping confidence intervals. Given the format limitations of the rebuttal period, we are unable to include a revised version. However, we have taken the following steps to enhance clarity in the final version:\n\n1. We will display only one representative method from each category of active learning approaches (Random, LCMD, SBAL, and BAIT) as baselines. The performance of all other methods is comprehensively reported in Tables E1–E4 in Appendix E. We will also carefully select a color scheme with improved perceptual separability.\n2. To have a clearer comparison between methods equipped with and without our PaPQS strategy, we will use distinct line styles (solid vs. dashed) and marker types (circles vs. diamonds) to differentiate their performance more effectively.\n3. We will change the confidence intervals from 95% to 90%. This will narrow the shaded regions and make the differences between methods clearer, especially in current densely populated plots. Importantly, this change does not affect the mean RMSE trends or alter any of the conclusions.\n\n> Question 2: Where are the tables (numerical results)? It is mentioned about Table E1, E2? \n\nThe appendix containing Tables E1–E4 was uploaded separately as part of the supplementary materials (within the ZIP file). This is also the reason why the hyperlinks in the main paper are currently inactive. In the final version, we will integrate the appendix into the main paper and ensure that all internal references and hyperlinks work as intended.\n\n> Question 3:  Fig 3(a) I don't see too much benefit over random.\n> \n\nWe agree that the improvements in Figure 3(a) may visually appear limited. However, the numerical results (RMSE reported in the table below) behind the figure do demonstrate that PaPQS achieves consistent improvements across all model architectures, with average improvements of **8.20%** for FNO, **5.54%** for SineNet, and **2.97%** for U-Net. The relatively smaller gain for U-Net may be attributed to its capability on the CNS task, as noted in a prior study of active learning on neural PDE solvers [1]. Thus, these results confirm that PaPQS provides consistent and significant improvements and  even when used with diverse model architectures.\n\n| Methods | FNO | FNO+PaPQS | SineNet | SineNet+PaPQS | U-Net | U-Net+PaPQS |\n| --- | --- | --- | --- | --- | --- | --- |\n| Iteration 0 | 3.900 | 3.900 | 2.800 | 2.800 | 2.660 | 2.660 |\n| Iteration 1 | 3.281 | 3.156 | 2.300 | 2.200 | 2.160 | 2.100 |\n| Iteration 2 | 3.046 | 2.786 | 1.860 | 1.760 | 1.856 | 1.780 |\n| Iteration 3 | 2.708 | 2.376 | 1.579 | 1.470 | 1.572 | 1.540 |\n\n\n> Question 4: Fig 4(b) Importance of the diversity component is not prominent but it has been claimed otherwise.\n\nWe understand that the effect of diversity regularization may appear subtle at a glance in Figure 4(b), especially in latter iterations. However, the numerical results show that incorporating the diversity term reduces RMSE during the earlier stages, where active learning query selection is the most critical. For instance, the RMSE at 512 samples decreases from 1.454 (without diversity) to 1.316 (with diversity), and at 1024 samples from 0.920 to 0.808, corresponding to reductions of approximately **9.5%** and **12.2%**, respectively. These early-stage gains are critical in low-resource regimes where each query must be highly informative, and they support our claim regarding the importance of the diversity term in improving sample efficiency. Furthermore, the distribution of selected samples (bottom panel of Fig. 4b) shows that diversity regularization promotes broader coverage across the viscosity domain, avoiding overconcentration in a tiny high-uncertainty region.\n\n> Question 5: Fig 4(a) The main EIG component also doesn't seem to be independent on it's own. \n\nWe have done experiments for the ablation analysis of the EIG component. The table below presents the RMSE results under different variants:\n\n- **Random**: Uniform random sampling from the initial condition space, with no acquisition mechanism.\n- **PaPQS w/o EIG**: Optimization is driven solely by diversity regularization (Eq. 8); updates are accepted based on entropy increase, without estimating EIG.\n- **PaPQS w EIG only**: The EIG component is retained and used to optimize initial conditions. Diversity term is removed.\n- **PaPQS**: Our full method.\n\n| Method | Iteration 0 | Iteration 1 | Iteration 2 | Iteration 3 | Iteration 4 |\n| --- | --- | --- | --- | --- | --- |\n| Random | 3.684 ± 1.203 | 3.278 ± 2.107 | 1.607 ± 0.485 | 1.062 ± 0.614 | 0.552 ± 0.133 |\n| PaPQS w/o EIG | 3.684 ± 1.203 | 3.139 ± 2.003 | 1.612 ± 0.369 | 1.103 ± 0.494 | 0.548 ± 0.144 |\n| PaPQS with only EIG  | 3.684 ± 1.203 | 1.454 ± 0.763 | 0.920 ± 0.132 | 0.528 ± 0.092 | 0.368 ± 0.076 |\n| PaPQS | 3.684 ± 1.203 | 1.316± 0.712 | 0.808± 0.123 | 0.531 ± 0.112 | 0.358 ± 0.115 |\n\nAs shown in the table, PaPQS with only EIG significantly improves over Random and PaPQS w/o EIG variants, confirming that the EIG component alone plays an essential role in improving sample efficiency, even without other strategies.\n\n> Question 6: Fig 2. Is there a reason to only use PaPQS with certain AL method and not others? Where is the curve for PaPQS alone?\n\nWe clarify that the baseline *Random* indicates that we train our model with data points randomly sampled from the initial condition distribution. *Random+PaPQS* means we apply the PaPQS and start the gradient-based optimization from these purely random samples. Thus, *Random+PaPQS* represents the performance of PaPQS alone, without the involvement of other active learning strategies.\n\nIn Fig. 2, we selected representative methods from distinct categories to demonstrate the compatibility and effectiveness of PaPQS in diverse active learning settings. **Random+PaPQS** shows how PaPQS performs starting from the random initial points. **SBAL+PaPQS** shows how PaPQS performs starting from the uncertainty-based sampling. **LCMD+PaPQS** shows how PaPQS performs starting from feature-based sampling. These two categories of active learning strategies have been widely adopted in recent active learning literature. Moreover, SBAL and LCMD are both the best baselines in their respective categories. Due to space constraints and computational cost, we focused on these representative baselines rather than all possible combinations.\n\n> Limitation: I think the time complexity of the proposed approach should be discussed as it is expected to be a bottleneck when making an informed choice about the training data needed.\n\nWe acknowledge that PaPQS introduces additional computational overhead. In Fig. 3(d), we reported the running time of the simulation and each component of the proposed PaPQS. These results indicate that the additional computations introduced by PaPQS are negligible, adding less than **1/20** of the total runtime compared to the simulation cost of the PDE solver. Besides, Fig. 3(c) supports that our method achieves significantly lower RMSE within the same time budget, supporting its practical efficiency.\n\n[1] Musekamp, Daniel, et al. \"Active learning for neural PDE solvers.\" arXiv preprint arXiv:2408.01536 (2024)."}, {"review_id": "zzvYayyiAK", "reviewer": "Reviewer_vMpD", "summary": "The paper introduces PaPQS, an active learning framework for neural partial differential equation solvers. Unlike traditional pool-based methods that are restricted to specific surrogate types, PaPQS synthesizes new initial conditions directly from the continuous input space. This approach enables finer control over informativeness and diversity in the query batch, helping avoid redundancy and instability. The framework is plug-and-play and generalizes well across multiple surrogate architectures, as demonstrated by comprehensive experiments.", "questions": "- Can you compare PaPQS with non-active learning pipelines that share the same objective goal with your paper. For example, the methods that aim for faster convergence, such as those leveraging pretraining or transfer learning?", "limitations": "The authors bring up the limitations in current design choices, such as the use of fixed step sizes and the absence of physics-informed or temporally adaptive strategies.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The plug-and-play design is easy to integrate and broadly applicable across different neural PDE surrogates, including U-Net, SineNet, and FNO.\n- The author provides systematic and well-controlled experiments that cover key aspects of the active learning pipeline, including efficiency, architecture generalizability, and data reusability.\n\nWeaknesses\n- The method requires access to the ground-truth PDE solver, making it inapplicable in settings where only precomputed or real-world datasets are available without simulation capability.\n- The performance gains are modest in several cases; for example, RMSE improvements over SBAL are minimal on Burgers, 2D-CNS, and CE equations. This experiment results cannot prove the efficacy of the proposed PaPQS method.", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your thoughtful questions and feedback.\n\n> The method requires access to the ground-truth PDE solver, making it inapplicable in settings where only precomputed or real-world datasets are available without simulation capability.\n\nOur method indeed assumes access to a ground-truth PDE solver, which allows evaluating arbitrary initial conditions in a continuous input space. This is crucial for our query synthesis framework, which actively optimizes over this space to find informative PDE settings. In contrast, precomputed or real-world datasets typically provide only a *fixed, finite* set of inputs, making it infeasible to apply our optimization-based approach directly. Therefore, our current method is not directly suited for such settings without constructing the corresponding pool-based active learning strategies as we explained in our responses to Reviewer kYvd. We would like to emphasize here that our motivation of developing PaPQS is to train neural PDE solvers that can accelerate the underlying PDE solutions with all possible parameters, boundary, and initial conditions in the PDE systems under study. This setup aligns with the nature of many PDE-based scientific modeling tasks, where the input space is inherently continuous and query synthesis is more appropriate than commonly adopted pool-based active learning. \n\n> The performance gains are modest in several cases; for example, RMSE improvements over SBAL are minimal on Burgers, 2D-CNS, and CE equations. This experiment results cannot prove the efficacy of the proposed PaPQS method.\n\nWhile we acknowledge that the RMSE improvements of PaPQS over SBAL are relatively modest on specific tasks such as Burgers', CE, and 2D-CNS PDE systems, we would like to emphasize the following:\n\n1. Even in PDE benchmarks where SBAL already performs strongly, PaPQS consistently achieves additional improvement (+12.48% on Burgers, +10.26% on CE, +6.35% on KS, and +3.90% on 2D-CNS). This trend demonstrates the robustness and generality of PaPQS across different model architectures and domains. PaPQS, as a plug-in refinement module that operates on top of existing acquisition methods, has its value as a general enhancement strategy.\n2. Current implementation uses a lightweight neural critic (a shallow CNN) and a limited number of gradient descent steps without sophisticated architectural design or algorithmic tuning. Despite this simplicity, PaPQS still delivers consistent performance gains. This suggests that the effectiveness of PaPQS stems from the generality of its framework, rather than from task-specific heuristics or finely tuned components. We believe this generality provides a strong foundation. Future work can further boost performance by exploring more expressive and efficient critic networks (e.g., Fourier Neural Operator), adaptive gradient schemes, or solver-aware inductive biases without altering the overall methodology.\n\n> Can you compare PaPQS with non-active learning pipelines that share the same objective goal with your paper. For example, the methods that aim for faster convergence, such as those leveraging pretraining or transfer learning?\n\nWe agree that PaPQS shares a common high-level goal with methods such as pretraining and transfer learning—that is, to accelerate convergence and reduce the sample complexity required for model training.\n\nHowever, PaPQS approaches this challenge from a different aspect, focusing on data acquisition rather than model initialization. Specifically, PaPQS actively synthesizes informative PDE initial conditions in a continuous design space. This enables informative data acquisition during training, which complements rather than replaces techniques like pretraining.\n\nIn contrast, pretraining or transfer learning methods typically rely on external data sources or related tasks. Their effectiveness depends on the similarity between pretraining and target domains and the quality of parameters finetuning. \n\nMoreover, PaPQS can be naturally combined with pretraining-based surrogates. For instance, one may start with a pretrained model and apply PaPQS to select the most valuable data points for fine-tuning, thereby improving sample efficiency and reducing redundant evaluations. This makes PaPQS particularly attractive in scenarios where full data acquisition is expensive, and selective fine-tuning on high-value samples is desirable. Exploring such hybrid pipelines is an exciting direction for future work."}, {"review_id": "RyYYqmm1Ne", "reviewer": "Reviewer_FANU", "summary": "The paper presents PaPQS, a query synthesis approach to sample datasets for neural PDE solvers. The method uses the mine-f method to estimate the expected information gain and SWA-Gaussian to sample the posterior distribution of model parameters. Combined with a distance-based metric for ensuring diversity, the acquisition function can be optimized using gradient ascent. The experiments show that the method can improve the samples generated by other active learning strategies.", "questions": "1. For combining SBAL and PaPQS, did you use EIG also as the acquisition function of SBAL or the original ensemble uncertainty?\n2. Why did you apply the critic T to individual states, and not the whole trajectory at once? Will this formulation still be able to consider dynamics in the information gain?\n3. Do you optimize the total number of samples to select in the given AL iteration all at once, or do you optimize within mini-batches?\nIn Eq. 8, did you use the distance in the initial parameter space and PDE parameters? If so, what distance measure did you use?\n4. Doesn’t the acceptance of only positive steps (Eq. 9) encourage getting stuck in local minima, which may have been prevented by the momentum of Adam otherwise?\n5. Minor details: In Eqs. 13 and 14, the summation index j seems not to be used within the sum. Misspelled “neural” in line 212.  In lines 133-136, you only mention batches of ICS, but shouldn’t it be batches of ICs and PDE parameters?", "limitations": "Only one sentence, which only mentions the performance on the 2D PDE. The modest performance on a single PDE does not show a general problem for specifically multi-field PDEs; it could also be the higher number of spatial dimensions or other reasons. For example, the strong dependence of the performance on the initialization could also be mentioned.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\n1. A novel method that can be added on top of every neural PDE solver and is faster than estimating uncertainty using ensembles.\n2. One of the rare works of AL that explores the query synthesis setting.\n3. Extensive experiments with confidence intervals.\n\nWeaknesses\n1. The method needs good initialization by a different AL method. If being optimized from scratch, it does not reach the performance of other AL strategies. \n2. The discussion of related work could be improved. For example, [1] and [2]  could be mentioned as applications of AL for neural PDE solvers.  In ln. 46-48: “feature-based acquisition methods emphasize input diversity but do not account for the relationship between PDE settings and model parameters” is not substantiated. For the first part, the feature-based acquisition function focuses on the coverage in feature-space and not in the inputs, and is also able to consider informativeness, for example [3]. Could you explain the second part about the relationship between PDE settings and model parameters? Don’t feature-based methods also use the gradient w.r.t. to the parameters as the feature representation [3]?  Lastly, PDERefiner (reference 11 in the paper)  is cited in ln. 34 as an example for AL, but this paper does not discuss the matter. \n\n[1] Li, S., Yu, X., Xing, W., Kirby, R., Narayan, A., & Zhe, S. (2024, April). Multi-resolution active learning of Fourier neural operators. In International Conference on Artificial Intelligence and Statistics (pp. 2440-2448). PMLR.  \n[2] Kim, Y., Kim, H., Ko, G., & Lee, J. Active Learning with Selective Time-Step Acquisition for PDEs. In Forty-second International Conference on Machine Learning.  \n[3] Holzmüller, D., Zaverkin, V., Kästner, J., & Steinwart, I. (2023). A framework and benchmark for deep batch active learning for regression. Journal of Machine Learning Research, 24(164), 1-81.", "was_revised": true, "final_justification": "I maintain my original rating of acceptance. The presented work is novel and can improve prior pool-based techniques. The authors responded well to the mentioned weaknesses and questions.  Other reviewers pointed out mainly issues with the clarity, which I think are, however, minor,  and can be addressed in the camera-ready version.  Additionally, I agree with the other reviewers that the improvement over the respective starting AL method is relatively small. However, the difference is pretty consistently positive across the investigated PDEs and dataset sizes, which is why it still can be considered relevant.", "rebuttal": "We appreciate your thoughtful questions and feedback.\n\n> Weakness: The discussion of related work could be improved. For example, [1] and [2] could be mentioned as applications of AL for neural PDE solvers. In ln. 46-48: “feature-based acquisition methods emphasize input diversity but do not account for the relationship between PDE settings and model parameters” is not substantiated. For the first part, the feature-based acquisition function focuses on the coverage in feature-space and not in the inputs, and is also able to consider informativeness, for example [3]. Could you explain the second part about the relationship between PDE settings and model parameters? Don’t feature-based methods also use the gradient w.r.t. to the parameters as the feature representation [3]? Lastly, PDERefiner (reference 11 in the paper) is cited in ln. 34 as an example for AL, but this paper does not discuss the matter.\n\nWe truly appreciate the reviewer's constructive suggestions as well as provided references. We will cite these references and provide the corresponding discussions in our revised introduction and background sections. \n\nAbout the first part, in l46-48, certain feature-based acquisition methods incorporate model parameters structurally by measuring in the embedding layer or introducing gradient kernels [3]. But we meant to explain that these methods only consider the relationships in the corresponding feature space implicitly, without explicitly modeling the mutual dependence between candidate inputs and model parameters, nor targeting the reduction of posterior uncertainty over parameters. We will revise the text to clarify our intention.\n\nAbout the second part,  [3] indeed incorporate model gradients with respect to parameters with $\\nabla_\\theta f(x)$ to define a similarity kernel, which allows them to measure a kind of sensitivity of predictions to model parameters and select diverse samples via strategies like MaxDiag, MaxDet, and LCMD optimization. However, our intended relationship between PDE settings and model pamaeters targets a direct posterior uncertainty reduction over model parameters. It is achieved by EIG and could be formulated by $\n\\text{EIG}(\\psi) = \\mathbb{E}_{p(u, \\theta \\mid \\psi)} \n\\left[ \n\\log \\frac{p(\\theta \\mid {u}, \\psi)}{p(\\theta)} \n\\right]$.\n\nIn other words, they use gradients of parameters as static features for diversity, while our method enables us to select settings that are more directly aligned with the goal of improving parameter estimation.\n\n[3] Holzmüller, D., Zaverkin, V., Kästner, J., & Steinwart, I. (2023). A framework and benchmark for deep batch active learning for regression. Journal of Machine Learning Research, 24(164), 1-81.\n\n> Question 1: For combining SBAL and PaPQS, did you use EIG also as the acquisition function of SBAL or the original ensemble uncertainty?\n\nIn the SBAL + PaPQS combination, we retain the original ensemble-based predictive variance as the acquisition function for SBAL. PaPQS is applied after the initial candidates are proposed by SBAL, serving as further continuous optimization that refines the candidate set by encouraging informativeness (via EIG) and diversity. This two-stage setup allows PaPQS to remain compatible with various upstream acquisition functions while improving their sample efficiency.\n\n> Question 2: Why did you apply the critic T to individual states, and not the whole trajectory at once? Will this formulation still be able to consider dynamics in the information gain?\n\nWe apply the critic to individual predicted states $u_t$ instead of the entire trajectory $u_{1:T}$ primarily for two reasons: (1) practical tractability considering high resolutions along the time dimension and (2) sufficient expressiveness of dynamics given the surrogate’s design.\n\nAlthough we apply the critic to individual predicted states $u_t$, the formulation still retains information about the full trajectory dynamics through two mechanisms. First, our surrogate model is autoregressive, where each state $u_t$ is predicted based on the previous state  $u_{t-1}$; this recursive structure ensures that the temporal dependencies and system dynamics are implicitly encoded in each step. \n\nSecond, the model input at each step includes the initial condition parameters of $u_0$, meaning the influence of the starting point propagates directly through the entire rollout and is embedded into each prediction. This propagation, carried through from the initial setting, is fully available to the neural critic. Together, these two designs ensure that the neural critic function can meaningfully reflect the informativeness over the trajectory, even though it is applied at the individual state.\n\n> Question 3: Do you optimize the total number of samples to select in the given AL iteration all at once, or do you optimize within mini-batches? In Eq. 8, did you use the distance in the initial parameter space and PDE parameters? If so, what distance measure did you use?\n\nWe use mini-batches, which we set to 64 for 2DCNS equations and 128 for others. \nWe compute distances in the initial condition parameter space. Take the Burgers’ equation as an example, the parameter space includes the viscosity coefficient, amplitude, wavenumber, and phase.\n For the inherently circular phase parameter, we adopt a squared circular distance defined as $\\left( \\min\\left( \\left| \\phi_1 - \\phi_2 \\right|, 2\\pi - \\left| \\phi_1 - \\phi_2 \\right| \\right) \\right)^2$.  We use the Euclidean distance for the other parameters. We also apply min–max normalization to each dimension based on its known bounds to ensure consistency across different parameters. \n\n> Question 4: Doesn’t the acceptance of only positive steps (Eq. 9) encourage getting stuck in local minima, which may have been prevented by the momentum of Adam otherwise?\n\nOur update criterion in Eq. (10) is not intended to replace optimizer dynamics such as momentum, but rather to combine EIG score and diversity in a tractable way. We clarify our rationale as follows:\n\n1. Non-differentiable diversity term: The diversity regularization $\\mathcal{H}(\\cdot)$ is kNN-based and non-differentiable, making it incompatible with gradient-based optimizers like Adam. To incoperate this with the EIG-based acquisition term $\\mathcal{A}(\\cdot)$, we adopt this filtering mechanism where only updates that may improve the joint objective (informativeness + diversity) are retained.\n2. As shown in our experiments (the table below), incorporating diversity via this hybrid policy improves performance in data-scarce regimes, suggesting that our approach effectively balances local improvement and global coverage.\n\n| Method         | iteration 0   | iteration 1   | iteration 2   |\n| -------------- | ------------- | ------------- | ------------- |\n| with diversity | 3.684 ± 1.203 | 1.316 ± 0.712 | 0.808 ± 0.123 |\n| w/o diversity  | 3.684 ± 1.203 | 1.454 ± 0.763 | 0.920 ± 0.132 |\n\n1. As we mentioned in line 271, the threshold $\\eta$ is not fixed but dynamically chosen as the median within the current batch. It means half updates are retained at each update step. This allows us to adaptively retain high-quality updates relative to the optimization landscape, rather than applying a static cutoff that could hinder exploration.\n\n> Question 5: Minor details: In Eqs. 13 and 14, the summation index j seems not to be used within the sum. Misspelled “neural” in line 212. In lines 133-136, you only mention batches of ICS, but shouldn’t it be batches of ICs and PDE parameters?\n\nThank you for pointing out these details. We will carefully check the typos and correct them in the final version.\n\nFor the Eqs. (13) and (14), we will correct the corresponding summation by the \n\n$$\\frac{e^{-1}}{N_{\\text{batch}}}\\sum_{j=1}^{N_{\\text{batch}}} e^{T_\\phi(\\theta_i, \\mathbf{u}_j(t, \\psi_j; \\theta_i))}$$ \nand \n\n$$ \\frac{e^{-1}}{N_{\\text{batch}}} \\sum_{j=1}^{N_{\\text{batch}}} \\nabla_{\\phi} e^{T_\\phi(\\theta_i, \\mathbf{u}_j(t, \\psi_j; \\theta_i))}$$, respectively.\n\nIn lines 133-136, it should be a batch of initial states $\\mathbf{u}^0$ and PDE parameters. We will clarify the components of the initial conditions (ICs) in the Background section of the final version."}, {"review_id": "YYAEu3j4iH", "reviewer": "Reviewer_kYvd", "summary": "The work addresses the learning of an autoregressive model for time series prediction in a context where data generation is expensive. The paper investigates learning tasks on physical time series as a representative example of this problem, where a simulator generates temporal trajectories with a natural trade-off between simulation quality and speed. To improve learning efficiency, the paper proposes an active learning approach for selecting initial conditions—not randomly, but in a way that ensures the resulting trajectories most benefit the model. \n\nSpecifically, the selected initial conditions are those expected to induce the greatest change in the current model. This approach is formalized within a probabilistic Bayesian framework. A key challenge lies in identifying, in advance and without time evolution, which initial conditions are likely to contribute the most information to the existing set of trajectories. To address this, a second model, referred to as the critic, is introduced. The critic does not require autoregressive rollout and can also provide estimates when the likelihood function of the autoregressive model is inaccessible. The entire system is jointly optimized, including the parameters of the autoregressive model, the initial conditions for new trajectories, and the critic’s parameters. The method is then evaluated through a series of experiments, including several differential equations, various active learning strategies, and different model architectures.", "questions": "- L57: A sentence in the paper ‘By explicitly targeting the reduction of the posterior uncertainty, our framework dynamically explores high-informative regions while avoiding instability-prone area’. Can you explain how to draw this conclusion? I wonder if it’s not exactly the opposite that’s the case. Shouldn’t an unstable trajectory maximize the expected information gain as processing an unstable trajectory would result in a significantly different model?\n\n- Are there any physics-specific components or assumptions in the method? If not, should the approach also be evaluated on non-physical time series tasks to better demonstrate its generality?\n\n- How does the method perform on stochastic time series?\n\nSuggestions:\n\n- As pointed out, a theoretical justification could really strengthen the paper. A study on a linear system (linear physics and linear model) may allow for a closed-form solution, which could reveal if the proposed method provides a lower variance estimator for the model parameters. \n\n- Another empirical test could be to visualize the optimization landscape. Most likely, the optimization landscape provided by the critic is smoother, especially in the beginning. Similarly, the spectrum of the Hessian indicates how fast gradient-based optimization finds solutions. This could be another strategy to gain some insights on the proposed method.\n\nMinor:\nL98: MSE loss, not RMSE\nL212: typo ‘neural’", "limitations": "Yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "I am uncertain about the novelty of the proposed approach and the use of critic networks in active learning. Critic networks are common in other areas to process information over time more effectively, such as reinforcement learning.\n\nMy main concern is that the overall setup seems overly complex, involving three nested non-convex optimization problems, each sensitive to initialization and hyperparameters. Even with extensive experimentation, I find it difficult to draw a conclusion about general applicability here.\n\nThe authors argue (lines 179–182) that small perturbations can lead to spikes in predictive variance due to the chaotic nature of the system and the autoregressive structure, thereby justifying their more complicated approach. But more generally, it is questionable if autoregressive setups or application of learning to chaotic system is a suitable task for approximate machine learning, because of that sensitivity. One could just as easily argue that the expected information gain is misled by small perturbations of a labeled initial condition that leads to a completely different trajectory and so drastically changes the model? \n\nI encourage the authors to explore theoretical approaches to better understand and justify their method. As noted, a purely empirical evaluation seems of limited value in this context. However, when combined with a solid theoretical analysis, a more compelling case could be made.\n\nStrengths\n\n- Relevant topic and direction of research\n\n- Many experiments and a detailed statistical evaluation\n\nWeaknesses:\n\n- Highly complex non-convex optimization\n\n- The introduction of the critic network feels abrupt—it appears in the middle of the technical derivation without sufficient context. As mentioned, the central challenge is to estimate the usefulness of a specific initial condition without evolving the system, and this is addressed using the critic. This connection should be communicated more clearly. Both the abstract and introduction should mention that a second neural network is involved and plays a key role in the method.\n\n- In my view, the derivation from the expected information gain (Equation 11) to the update equations (Equation 13) is difficult to follow. The text primarily describes what is being done, but the reasons behind the design choices are not sufficiently explained. Any additional intuition or explanation would be helpful here. It might also improve clarity to restructure Sections 3.1  (optimizer and loss term 2/regularization term)  and 3.2 (loss term 1/EIG and update equations). Also, devoting half a page to the Adam optimizer seems unnecessary, as it is a standard choice.\n\n- Equation 4, in my opinion, does not provide much insight, and the mathematical notation is somewhat confusing.\n\n- Regarding Claim 1 and Proposition 1, the statements are not mathematically precise. For example: “Assume that the dominant sources of epistemic uncertainty of the model can be captured by θ_out” is vague and would benefit from clarification. Additionally, the final step in the proof (Equation 18) should be explicitly referenced in the appendix as well. Since this step appears to be crucial, an intuitive explanation of the identity in the main text could help readers better understand the overall argument.", "was_revised": true, "final_justification": "Post rebuttal: my main concern is that we are dealing with three nested nonlinear optimization problems. These are commonly approached by iterative methods that utilize the current local structure of the problem, as you noted, and I do not question that this is appropriate. My issue, however, is that due to this complicated setup, it is difficult to compare methods that solve it and conclude that one is better than another, which you claim in your paper. For me, a purely experimental evaluation is not enough given this complex setup.\n\nA theoretical analysis, as provided in your references [2,3] (rebuttal not paper), captures one part, namely the active learning. However, there is still the iterative updating of the model and the critic. Assume linear function approximation for both of them, then you enter a regime where gradient descent is well studied and hence, where all three components are individually understood. My suggestion is to put these ideas together for your setup, where then 3 nested linear problems still form an overall linear problem. The way this multiplies out depends on the specifics of your method. If the final expression shows your method's advantage, then, combined with the experimental results, this will be a strong paper. \n\nI wouldn't oppose if others argue for acceptance, but I'm inclined to keep my current score of a weak reject.", "rebuttal": "We appreciate your thoughtful comments.\n\n>W1: Highly complex non-convex optimization & theoretical justification on linear system\n\nWe acknowledge that our acquisition optimization is a non-convex problem, which is a common and often unavoidable characteristic for active learning with complex models and in more general optimal experimental design under uncertainty, especially with high-dimensional and structured design spaces such as PDE initial condition space. The common solution strategies typically follow gradient-based methods, which is what we adopted with our proposed gradient estimates based on EIG acquisition function. We acknowledge that such an optimization solution may not lead to global optimal solutions. Nevertheless, our optimization converges reliably in practice with standard optimizers (e.g., Adam) at a moderate computational cost. Empirically, we observe consistent improvements across benchmarks (see Tables E1–E4), indicating that the local optima found by our method are sufficiently effective.\n\nTheoretical analysis of EIG-based methods and active learning for linear systems, indicated by the reviewer, has been studied in literature, including references [2][3].  If the model is linear, the scenario simplifies to a linear projection of the output, resulting in maximizing the determinant of identity plus the Hessian of the misfit. Consequently, in this case, the focus directly shifts to reducing uncertainty with respect to the model parameter covariance matrix.  \n\nHowever, theoretical analysis to extend to PDE neural solvers can be challenging as commented by the reviewer. Our work instead adopts a practical approach by approximating EIG using a neural critic trained with the MINE loss. This enables scalable acquisition in complex PDE settings, where theoretical guarantees are difficult to derive. We agree that extending theoretical analysis to more general nonlinear systems (e.g., constrained surrogates or kernel-based approximations) would be a valuable direction for future research, and we will discuss our limitations and this future direction in the final version.\n\n> W2: The introduction of the critic network feels abrupt.…\n\nWe will clarify in the abstract that our method uses a neural critic function to estimate the EIG.\n\nRevised version:\n\nTo improve the utility of candidate settings, PaPQS optimizes the Expected Information Gain (EIG) estimated by a learned neural critic for each PDE setting without evolving the PDE system, enabling model-aware exploration of the design space via backpropagation through the neural PDE solution trajectories. Meanwhile, it encourages batch diversity to prevent sampled PDE settings from overconcentrating in narrow regions of the input space.\n\nWe will also supplement the relative description in the introduction in the final version.\n\n> W3: The derivation problem in Sec. 3.2. Devoting half a page to the Adam optimizer seems unnecessary.\n\nSection 3.2 follows the following organization: acquisition objective - required sampling - optimization procedure. Firstly, we introduce the acquisition function based on EIG. We then describe how we estimate EIG using the MINE-f lower bound (Eq. 11), followed by posterior sampling via SWAG (Eq. 12) to enable sampling of $\\theta$, which are required for the bound computation. Finally, we provide the gradient formulation for optimizing the initial conditions $\\psi$ through gradient ascent (Eq. 13), including how we backpropagate through the neural PDE solver. \n\nTo improve clarity, we will better connect our design choices and strengthen the intuition for our gradient-based acquisition process. We will shorten the Adam part and instead use the space to elaborate on the motivations behind our EIG formulation and its differentiability, which we believe will significantly improve the readability and justification of our method.\n\n> W4: Eq. 4 does not provide much insight, and the mathematical notation is somewhat confusing.\n\nOur intention with this formulation was to concisely summarize the query synthesis procedure as a gradient-based optimization over a batch of initial conditions, which is guided by the informativeness scoring function $\\pi(\\cdot)$. \n\nTo improve clarity, we will: (1) update the notation by removing the argmax over $\\mathcal{G}$ to avoid the confusion with misled impression considering a search over operators. Instead, we will present it as a regular optimization procedure; (2) emphasize that the scoring function $\\pi(\\cdot)$ is explicitly defined in Eq. (9), where its components are detailed (We will also add the corresponding hyperlink); (3) provide a more intuitive explanation in the text to help readers better understand the role of Eq (4) as a summary of the iterative query synthesis process via backpropagation through both the surrogate model and the learned neural critic. \n\n> W5: Regarding Claim 1 and Proposition 1, the statements are not mathematically precise. Eq.18 should be explicitly referenced...\n\nWe will explicitly reference Eq. 18 in the main text and clarify its connection to the MINE-f lower bound of mutual information. We will revise Proposition 1 as follows:\n\n...Assume that uncertainty over surrogate predictions $\\mathbf{u}$  is dominated by the output layer parameters $\\theta_{out}$ , i.e., $p(\\mathbf{u}|\\theta,\\psi)= p(\\mathbf{u}|\\theta_{out},\\psi)$. Then, the $EIG(\\psi)$ can be approximated by sampling only  $\\theta_{out}$, i.e., .$EIG(\\psi)=I(\\theta;\\mathbf{u}|\\psi)=I(\\theta_{out};\\mathbf{u}|\\psi)$.  A valid lower bound...\n\nWe will introduce mathematical descriptions for the terms in Claim 1:\n\nprediction uncertainty: $Var_{\\theta \\sim p(\\theta)}[\\mathbb{E}[\\mathbf{u} | \\theta,\\psi]]$\n\n> Q1: L57: A sentence in the paper ‘By explicitly...instability-prone area’. Explain how to draw this conclusion...\n\n> One could just as easily argue that the EIG is misled by small perturbations of a labeled initial condition that leads to a completely different trajectory and so drastically changes the model?\n\nWe agree that in chaotic systems, small perturbations can cause large variations in model outputs, which can in turn yield high predictive variance or even numerically large EIG values. However, not all this kind of uncertainty is epistemically informative. For these instances, we need to determine whether the observed uncertainty stems from a lack of knowledge about the model parameters (epistemic) or from intrinsic instability that cannot be reduced through learning (aleatoric or chaotic).\n\nOur framework solves this problem by using EIG as the acquisition objective, which explicitly aims to reduce posterior uncertainty over model parameters. This objective steers the acquisition process toward epistemically informative and learnable regions, rather than those where uncertainty primarily arises from trajectory instability. For example, in neural PDE surrogate modeling, certain initial conditions can lead to sensitive or chaotic responses, but such regions may inflate entropy without contributing useful gradients for learning. Training on these unstable trajectories may also hurt model robustness and generalization. \n\nIn a word, compared to previous variance-based acquisition methods, our EIG-based approach inherently conditions on both observations and model parameters, capturing mutual informativeness. This allows PaPQS to mitigate the risk of being misled by unstable trajectories that often reflect high variance but low epistemic gain.\n\nWe will revise the related presentation to make our motivation clearer. We hope this addresses the reviewer’s concern.\n\n> Q2&3: Are there physics-specific components or assumptions? Should the approach also be evaluated on non-physical time series? How does the method perform on stochastic time series?\n\nWhile PaPQS does not rely on any physics-specific priors or inductive biases, our current design focuses on PDE solvers and continuous initial condition spaces. This aligns with PDE-based scientific modeling, where the design space is inherently continuous and query synthesis is preferable to standard pool-based methods. This motivates the development of PaPQS as the focus of active learning is to train neural PDE solvers that can accelerate the underlying PDE solutions with all possible parameters, boundary and initial conditions in the PDE systems under study. \n\nFor the review suggested, when developing active learning with non-physical time series, typical active learning benchmarks rely on fixed, pre-collected datasets and lack a natural continuous query space, and therefore most of benchmarked active learning methods are pool-based instead of considering continuous optimization as in PaPQS. This limits the direct applicability of our method in its current form. Extending PaPQS to such domains would require rethinking the query synthesis mechanism, potentially by constructing a corresponding pool-based variant, for which our developed gradient estimates, one of our methodological contributions in PaPQS, may not be needed as the active learning can be done by either ranking or greedy search based on estimated EIG. \n\nRegarding stochastic time series, our method can still be applicable, especially for those with a surrogate model that explicitly outputs a predictive distribution. Since PaPQS estimates EIG, access to the predictive likelihood enables the use of techniques such as nested Monte Carlo or contrastive methods [1] to approximate the EIG more directly. We believe this extension is promising, especially when the surrogate is probabilistic and uncertainty-aware.\n\n[1] Foster, Adam, et al. \"A unified stochastic gradient approach to designing Bayesian-optimal experiments.\" AISTATS. PMLR, 2020.\n\n[2] Alexanderian, et al. \"On Bayesian A-and D-optimal experimental designs in infinite dimensions.\" (2016):671-695.\n\n[3] Attia, Ahmed, et al. \"Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems.\" Inverse Problems,2018:095009."}]

Author rebuttal:

Meta-review: This paper proposes PaPQS, a plug-and-play query synthesis active learning framework for neural PDE solvers. The method is novel in that it synthesizes informative PDE initial conditions directly in the continuous design space, guided by a neural critic estimating expected information gain (EIG) and a diversity regularizer. The idea is original and timely, given the growing interest in reducing simulation costs for training neural PDE surrogates.

The reviewers highlight several strengths: PaPQS is general and can be applied to multiple surrogate architectures and acquisition strategies, it consistently improves over baselines, and it introduces query synthesis for PDE settings, which has not been systematically explored before. The experiments are extensive and include ablation studies.

Concerns were raised regarding clarity of presentation, modest performance gains over strong baselines, and the complexity of the nested optimization setup. One reviewer requested a stronger theoretical justification, noting the difficulty of drawing conclusions from purely empirical evidence. The authors addressed these points in rebuttal, committing to improved presentation and providing a useful linear-case analysis clarifying why EIG avoids noise inflation seen in variance-based methods.

While some improvements are modest, the gains are consistent across PDE systems and architectures, and the method is complementary to existing strategies rather than a replacement. The reliance on a ground-truth PDE solver does limit applicability, but the setup remains relevant for scientific machine learning contexts where simulation is available.

On balance, the novelty of the approach, the broad applicability, and the consistent improvements outweigh the weaknesses. I recommend acceptance.
Decision: Accept (poster)