PhySense: Sensor Placement Optimization for Accurate Physics Sensing
Authors: Yuezhou Ma
Haixu Wu
Hang Zhou
Huikun Weng
Jianmin Wang
Mingsheng Long
Venue: NeurIPS 2025 oral ()
Keywords: Physics sensing
sensor placement
flow models
Primary area: deep_learning
TL;DR: We propose a synergistic two-stage framework that learns to jointly reconstruct physical fields and to optimize sensor placements with theoretical guarantees, both aiming for accurate physics sensing.

Abstract: Physics sensing plays a central role in many scientific and engineering domains, which inherently involves two coupled tasks: reconstructing dense physical fields from sparse observations and optimizing scattered sensor placements to observe maximum information. While deep learning has made rapid advances in sparse-data reconstruction, existing methods generally omit optimization of sensor placements, leaving the mutual enhancement between reconstruction and placement on the shelf. To change this suboptimal practice, we propose PhySense, a synergistic two-stage framework that learns to jointly reconstruct physical fields and to optimize sensor placements, both aiming for accurate physics sensing. The first stage involves a flow-based generative model enhanced by cross-attention to adaptively fuse sparse observations. Leveraging the reconstruction feedback, the second stage performs sensor placement via projected gradient descent to satisfy spatial constraints. We further prove that the learning objectives of the two stages are consistent with classical variance-minimization principles, providing theoretical guarantees. Extensive experiments across three challenging benchmarks, especially a 3D geometry dataset, indicate PhySense achieves state-of-the-art physics sensing accuracy and discovers informative sensor placements previously unconsidered. Code is available at this repository: https://github.com/thuml/PhySense.

PhySense: Sensor Placement Optimization for Accurate Physics Sensing
Yuezhou Ma, Haixu Wu, Hang Zhou, Huikun Weng, Jianmin Wang, Mingsheng Long
School of Software, BNRist, Tsinghua University, China
1 com, (zhou-h23, senghk22))0maiin tsinghua edu. cn.
{j imwaang mingghhnnghttsinnguua edu.cn
Abstract
Physics sensing plays central role in many scientific and engineering domains, which inherently involves two coupled tasks: reconstructing dense physical fields from sparse observations and optimizing scattered sensor placements observe maximum information. While deep learning has made rapid advances in sparse-date data reconstruction, existing methods generally omit optimization of sensor placements, leaving the mutual enhancement between reconstruction and placement on the shelf. To change this suboptimal practice, propose PhySense, synergistic two- stage framework that learns to jointly reconstruct physical fields and to optimize sensor placements, both aiming for accurate physics sensing. The first stage involves a flow-based generative model enhanced by cross-attention to adaptively fuse sparse observations. Leveraging the reconstruction feedback, the second stage performs sensor placement via projected gradient descent to satisfy spatial constraints. We further prove that the learning objectives of the two stages are consistent with classical variance-minie minimization principles, providing theoretical guarantees. Extensive experiments across three challenging benchmarks, especially 3D geometry dataset, indicate PhySense achieves state-of-the-ar physics sensing accuracy and discovers informative sensor placements previously unconsidered. Code is available at this repository: https//gihha.cco//bum//////.s
Introduction
Physics sensing remains foundational task across several domains 45
23 47
including fluid
dynamics [2 15] 50), meteorology [39 5], and industrial applications 11 37. The task aims to reconstruct the spatiotemporal information of a physical system from limited sparse observations gathered by pre-placed sensors. real-world scenarios, the number of sensors is usually limited due to spatial constraints, power consumption, and environmental restrictions. As result, allocating the limited sensors to the most informative positions is crucial for high reconstruction accuracy. Poorly placed sensors lead to spatial blind spots and information loss, resulting in insufficient observations, while well-placed sensors enable the recovery of fine-scale physical patterns. Therefore, advancing accurate physics sensing requires not only improving the capabilities of reconstruction models, but also developing principled strategies to determine optimal sensor placements.
Recent advances in deep learning have demonstrated remarkable capabilities in sparse-data recon- struction, owing to its ability to approximate highly nonlinear mappings [16
44 37 between scattered observations and dense physical fields. Current reconstruction methods fall into two categories: deter- ministic and generative. Deterministic methods such as VoronoiCNN 14 employ Voronoi-tessellated grids for CNN-based reconstruction, while Senseiver [41] utilizes implicit neural representations [19] to establish correlations between sparse measurements and query points. Morcover, due to the inher- ently ill iooposed nature 2 of the reconstruction task, where sparse observations provide insufficient
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Optimized Placement
Sensor position
Reconstruction
Error
Sensor position
Reconstruction
Error
Figure I: Performance comparison under same reconstruction model but different sensor placements. (a) Random placement yields poor results due inadequate spatial coverage. (b) Our optimized placement achieves accurate reconstruction by discovering informative regions, including side mirrors.
information for recovering dense fields, generative models 17 explored. For example, DiffusionPDE [88] and GM 228 model physical field distributions via diffusion models, and integrate sensor observations during sampling through training-free guid- ance 7 49]. Despite their reconstruction accuracy, these methods share fundamental limitation: their practical experiments, sensor placements are either randomly distributed or arbitrarily fixed rather than being systematically optimized. Consequently, the synergistic potential of co-optimizing reconstruction quality and placement strategy remains largely unexplored.
42 30 29]
have been increasingly
In practice, sensor placement determines the structure of the observation space and constrains what the model can perceive, while the feedback from model's reconstruction should guide where sensing most beneficial. Notably, as illustrated in Figure] | even under the identical reconstruction model, the performance varies dramatically with different sensor placements. Random sensor placement fails to capture important spatial regions, resulting in degraded performance, whereas placements strategically optimized through reconstruction feedback yield significantly enhanced reconstruction accuracy by targeting informative regions, such as side mirrors of car. These results suggest the existence of positive feedback loop between reconstruction quality and sensor placement optimization.
Building upon these insights, this paper presents PhySense, a synergistic two-stage framework that combines physical field reconstruction with sensor placement optimization. In the first stage, we train a base reconstruction model using flow matching [29] capable of generating dense fields from all feasible sensor placements. In the second stage, leveraging the reconstruction feedback, sensors optimized using projected gradient descent to satisfy spatial geometries. Furthermore, we prove that the learning objectives of the stages are consistent with classical pariance-minimian providing theoretical guarantees. We evaluate our method across three challenging benchmarks, including turbulent flow simulations, reanalysis of global sea temperature, and industrial simulations of aerodynamic surface pressure over a 3D car on irregular geometry. In all cases, PhySense consistently achieves state-of-the-art performance and discovers some informative sensor placements previously unconsidered. Overall, our contributions can be summarized as follows:
principles,
We introduce PhySense, synergistic two-stage framework for accurate physics sensing, which integrates flow-based generative reconstruction model with sensor placement optimization strategy through projected gradient descent to respect spatial constraints.
We prove the learning objectives of reconstruction model and sensor placement optimization are consistent with classical crrianc--miniiization targets, providing theoretical guarantees.
PhySense achieves consistent state-of-the reconstruction accuracy with 49% relative gain across three challenging benchmarks and discovers informative sensor placements.
2 Preliminaries
2.1 Deep Reconstruction Models
Recent deep learning methods have demonstrated promising performance in reconstructing dense physical fields from sparse measurements. Existing reconstruction models can be broadly cate- gorized into deterministic and generative approaches. Representative deterministic methods are VoronoiCNN [4] and Senseiver [41. VoronoiCNN constructs a Voronoi tessellation from scattered sensor observations, and learns to map the resulting structured representation to the target physical field. Senseiver encodes arbitrarily sized scattered inputs into latent space using cross-attention and adopts implicit neural representation its decoder. However, since sparse reconstruction is

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inherently ill- ill-posed problem, these deterministic methods lack the ability to characterize uncertainty in the reconstructed fields. Deep generative models have also been applied the sparse reconstruction task. For example, S [28] and DiffusionPDE [18] both tackle the reconstruction task by first learning a generative model of the underlying dynamics and then guiding the sampling process using sparse sensor observations. Crucially, these methods only utilize sparse observations during the sampling process not during training-tte thus requiring thousands of sampling steps to align with sparse imputs, which compromises the efficiency of deep models. Despite promising reconstruction accuracy, all their experiments are conducted under random or fixed sensor placements, which leads to suboptimal reconstruction accuracy due to potentially non- non-informative sensor placements.
2.2 Sensor Placement Optimization
Optimal criterion Optimal sensor placement aims to maximize information extraction from physical systems by strategically positioning sensors. This problem rooted optimal experimental design (OED) theory (3]/3,, which provides basic criteria for evaluating sensor placements. Three most widely-used criteria are: (1) A-optimality, which minimizes the average variance of all data estimates; (2) D-optimality, which maximizes the overall information content; and (3) E-optimality, which focuses on the worst-case estimation error. More details about these criteria can be found in [32]. In this work, we primarily focus on A-optimality. While these criteria originated in classical OED, they have been adapted to sensor placement optimization by incorporating spatial constraints [26].
Optimization algorithm Sensor placement optimization a fundamentally hard problem due to its inherent high-dimensional nature. For finite option problems, an exhaustive search over all feasible placements may be tractable, but for more common high nigh-dimensional problems, principled approximations become necessary. The theories of compressed sensing [ show that signals admitting sparse representations in a known basis, such as the Fourier basis, can be accurately reconstructed with small number of sensor observations. However, in real-world applications, the assumption of a suitable and known basis may not hold. Moreover, recent advances in machine learning have enabled data-driven discovery of oow-dimensional structures such low-rank subspaces [1] 46 27).
This shift from fixed to data-driven representation spaces has spurred the development of placement optimization methods that leverage empirical priors learned latent bases for efficient compression and reconstruction [32 40). For example, the SSPOR algorithm (32 first learns a data-driven basis, then applies QR factorization select basis-specific sensor locations that maximize reconstruction performance. While recent work has begun to explore catistical-basee sensor placement, deep lcarning-basee based approaches remain under explored. Besides, the further challenge lies in how to effectively co-optimize the reconstruction and the placement for accurate physics sensing.
2.3 Flow-based Models
Flow matching
29
30
31
is a framework for solving transport mapping problem: given two
distributions X To and X Ti, the goal is to find a continuous mapping I', such that 7 T'(x)) ~
TT. The method's efficiency and capacity have made it particularly valuable in vision tasks [[]]. Unlike diffusion models that require computationally intensive ODE (42 SDE [7 43 24 solving, flow matching introduces an ODE model that transfers No to T optimally via the straight line, theoretically the shortest path between two points, dX, = va (X,, t) dt, t E [0, 1], where vo (X,, f) a learnable velocity field. To estimate v. flow matching solves simple least squares regression problem that fits va to (X, , Xo). The resulting dynamics generate samples via efficient push- forward along the theoretically shortest path, X X, + vo(X,, ()dI, where X, ~ To- this paper, we consider to integrate this advanced technique for more efficient reconstruction model.
3 Method
To enable accurate sensing of physical fields, we co-optimize reconstruction and placement through a synergistic two-stage framework. Our approach begins by training a flow-based model capable of reconstructing dense fields from scattered measurements across all feasible placements. Leveraging the reconstruction feedback. PhySense seamlessly integrates reconstruction and placement processes, forming closed loop (Fig. (a)), and formulates placement as spatially constrained optimization problem, efficiently solved via projected gradient descent. Moreover, the optimization objective theoretically provides a bidirectional control with respect to the variance-minn minimization target.
3

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Physics Sensing Paradigm Reconstruction
Flow-based Reconstructor
Nolay Field X
X
Croas Attentionn
Feed Seraaad
PhySense
Velocity
Optimal Velocity X
Sensor Observations
Sensor
Reconstruction Evaluation
Placement
Reconstruction Model
X Led
Reconstructor
Reconstructor Feedbback
Backpropagation-
Figure 2: (a) PhySense works as a closed-loop physics sensing paradigm that iteratively co-optimizes sensor placements and reconstruction quality under the reconstruction feedback. (b) Reconstruction stage: A flow-based model with cross-attention mechanisms trained to process arbitrary sensor placements ('t--" indicates information flow for subsequent generation and placement optimization).
Let D C Ied be a bounded domain, where exists C physical fields of interest. We consider M {p1..... pm sensors distributed across D, where each sensor measures the target quantities at its position. Our physics sensing task is to accurately reconstruct the entire physical field X1 based on sparse sensor observations X1 (p") under the optimal sensor placement p".
Problem setup
3.1 Flow-based Reconstructor
Physics sensing, as an inverse problem to reconstruct dense physical fields from sparse observations, naturally requires generative approaches capable of handling the inherent ill-posedsees. In our framework, we formulate the reconstruction task as an optimal transport flow problem between the target physical field distribution и and standard Gaussian source distribution no. with sensor placements and sparse observations serving as conditioning variables.
Notably, the reconstruction model is expected to provide instructive feedback for any current sensor placement in the subsequent placement optimization stage. Therefore, we implement random sensor placement sampling during the flow model training, which helps the model extract maximal useful information from any feasible sensor placement. Therefore, our training flow loss is formalized as
L f- ExxXX.P [[(X, -x)) vo (P))) di = + t)xo, (1)
where vo represents our learned velocity model, and p denotes a sensor placement randomly sampled from a uniform distribution on D. This model design must address two practical challenges: (1) handling diverse physical domains spanning regular grids irregular meshes, and (2) effectively incorporating sparse observations for consistent reconstruction. For the first challenge, we leverage the state-of-the-arr DiT [35] [0] for regular grids and Transolver (8). the most advanced approach for general geometry modeling, for irregular geometries, ensuring optimal parameterization of the velocity field across various domains. The second challenge resolved through cross- -ooss-attnntion mechanism that dynamically associates scattered sensor measurements with their relevant spatial influence regions during training. This only provides some physical insights into sensor-field interactions but also enhances generalization across varying number and placements of sensors.
Theoretically, prove that reconstructor learn the conditional expectation of physical fields. Theorem 3.1 (Flow- reconstructor is an unbiased estimator of physical fields). Let the training flow loss Eq.
[ be minimized over class of velocity fields v. The optimal learned velocity, v va, t, p. (i))) = X - X Xt, t, P, (1))
equals the conditional mean of all feasible directions between the target data X, and the initial noise Xo. Define the reconstructed data as the integral along the optimal flow, X1 == Xo - So (X,,, L, D, (p)) dt, then X, is unbiased estimator of the target physical fields given sensor placement p and corresponding observation X1(p), i.e. E P. X,(p)] E[X,, p. (p)).
3.2 Sensor Placement Optimization
Based on the well-trained reconstructor, we formalize placement optimization as constrained optimization problem efficiently solved through newly proposed projected gradient descent strategy to respect spatial constraints, yielding near A-optimal [33 placements with theoretical guarantees.

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Projected gradient descent Optimizing sensor place- ment presents unique challenges compared standard model parameter optimization, since solutions must adhere to complex spatial constraints imposed by the physical domain D. While gradient-base updates can suggest improved placements, naive application often violates geometric
Noisy
X
Field
Well-Trained Reconstructor
Velocity Vt
Sensor Obs
Minimizing Flow Loss
Backpropagation
in irregular geometries. Therefore, we employ projected gradi- descent (Fig. 3). which iteratively (1) computes accuracy accuracy-improving gradients and (2) projects the up- dated placements back to the feasible space via nearest- point mapping. This process can be formalized as
Projected Gradient
Figure 3: Placement optimization stage: Based well-trained reconstructor, sen- sors are optimized by coometry-constralne projected gradient descent minimize the flow loss with theoretical guarantees.
p (k++1 = Projz (pth)
v/ Phtow (p (*)))
(2)
where p!* denotes the sensor placement at the k-th iteration, and (-)j denotes the projection operator onto the feasible domain D. The strategy maintains strict spatial constraints while systematically exploring superior placements, effectively bridging unconstrained optimization with physical feasibility requirements.
Optimization objective The selection of optimization objective significantly impacts both compu- tational cost and reconstruction accuracy. Although minimizing the reconstruction error || X,| seems conceptually straightforward, its requirement for backpropagation among multiple flow-process steps results in prohibitive computational and memory overhead. In contrast, the flow loss defined in Eq. ① maintains computational tractability through single flow-process step gradient calculation and, due to the optimal transport path between Xo and X1, theoretically guarantees near A-optimal sensor placements, dual advantage that makes our preferred choice.
The following theoretical analysis elucidates what our placement optimization objective fundamen- tally captures and how it relates to the rell-established A-optimality framework.
Assumption 3.2 (Conditional Gaussian assumption). We assume the target distribution is condi- tionally Gaussian, specifically, X₁ (p. Xi())) (O, Ep), where Ep the covariance matrix determined by the sensor placement p. Furthermore, Xo and X, are mutually independent.
Definition 3.3 (Classical A-optimal placement). We define the conditional covariance matrix of X given placement and corresponding observation X (p) is Var(p) = [[[(X -EXX, (X,,))XX,EX,, P. pp))(p)] = Ep. sensor placement p is called A-optimal if minimizes the trace of Ep.
p A-ape arg min CA(P), with (A(P) tr(Ep) E E -0,,EX111!
EX,1²
PE
where Tm denotes all feasible placements for sensors. From definition, A A-optimal placement attains the minimal total conditional variance of the reconstructed field among feasible placements. Definition 3.4 miniiiize sensor placement). Assume the velocity model attains its optimum v Another optimal placement defined if it minimizes the residual flow loss P* arg mim Lnow (p), where Lnow := Ex., (())) [=]]] [[²)
which measures the conditional variance when the optimal transport flow path is used instead of the true data transition.
Theorem 3.5 (The objectives of two optimal placements are mutually controlled). Under
sumption 3.2 the objectives of two optimal placements can be simplified to
CA(P) = Mp),, frow (p) = Σ 22 (wrP))
1=1
with eigenvalues {Ax(p)] Ep- Thus, two objectives control each other via following inequality:
/ can < CA(P) < n2 Chow (p).
Proof sketch. The proof's core lies the non-trivial simplification Lao through careful analysis of the structure of the conditional covariance matrix. Owing to the favorable properties of the optimal transport path, the involved complex integrals can be calculated closed form, offering further insight into the structure of the flow matching. More details can be found in Appendix A.2
D
5

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Remark 3.6. Consequently. the two objectives admit a two-sided polynomial bound of degree 2. which implies that minimizing Laow inherently constrains Lx to lie within a bounded neighborhood of its optimum, and vice versa. This provides a theoretical guarantee for achieving near A-optimal sensor placement under the efficiently implementable flow-loss- caa--loss-minimizattin objective.
Overall design In summary, PhySense seamlessly integrates two synergistic stages for accurate physics sensing. (1) Reconstruction: a flow-based model learns to reconstruct dense physical fields from arbitrary scattered inputs through optimal transport flow dynamics. During training, both the number and placements of sensors are randomized to ensure generalizability. (2) Placement optimization: under exed-number sensors, placements are optimized on the residual flow loss via projected gradient descent, constrained to feasible geometries (c.g., 3D surfaces). Theoretically, we prove that the reconstruction model learns the conditional expectation of target physical fields, and the flow-loss minimization objective polynomially controls the classical A-optimal criteria.
4 Experiments
We perform comprehensive experiments to evaluate PhySense, including turbulent flow simulation, reanalysis global sea temperature with land constraints and aerodynamic pressure on a 3D car.
Benchmarks As presented in Table I] our experiments encompass both regular grids and unstructured meshes in the 2D and 3D do- mains. The benchmarks include: (1) Turbu- lent Flow: numerical simulations of channel flow turbulence following the setup in Sen- seiver [41]; (2) Sea Temperature: 27-year daily sea surface temperature at 1 resolution
Table 1: Summary of benchmarks in PhySense.
#Mesh records the size of discretized mesh points.
Benchmarks
Geometry
#Dim #Mesh
Turbulent Flow
Regular Grid
2D
6144
Sea Temperature
Land Constraints
2D
64800
Car Aerodynamics
Unstructured Mesh
95428
with intricate land constraints from the GLORYS12 reanalysis dataset [2T]: and (3) Car Aerodynam- ies: high-fidelity OpenFOAM simulations 200 of surface pressure
complex car geometry from ShapeNet [6), where we reconstruct the irregular pressure fields from scattered sensor observations.
Baselines We evaluate PhySense via two approaches: (1) comparing the reconstruction model under randomly sampled sensor placements, and (2) comparing sensor placements with frozen PhySense's 's's reconstruction model. The reconstruction benchmarks include five representative basclines: classical statistical-basse method SSPOR (32), deterministic models VoronoiCNN [14] and Senseiver [41], and generative approaches SMGM (8] and DiffusionPDE [88. For placement evaluation, we compare optimized placement against random sampling and SSPOR's ' optimized placement 32
Implementations All methods of each benchmark maintain identical training configurations in- cluding epochs, batch sizes, ADAM optimizer (25), and maximum learning rate, while preserving other details as specified in their original papers. For the reconstruction model, we employ DiT [35] for turbulent flow and sea temperature benchmarks, while using Transolver [48) as a Transformer backbone to fit irregular mesh for car acrodynamies benchmark. The placement optimization stage runs for 5 epochs with the frozen reconstruction model. Both two stages employ relative L2 as the metrics for the reconstructed fields. Additional implementation details are provided in Appendix C
4.1 Turbulent Flow
Setups The benchmark focuses on reconstructing turbulent flow within a channel using 2-D slice discretized on a 128 x 48 regular grid. This dataset widely used, with classical models such as Senseiver [4] and VoroniCNN [14] serving as benchmarks. We replicate Senseiver's 's's setup; specifically, 25 300 sensors are randomly selected train the reconstructor, with sensor positions uniformly sampled across the entire domain. A setting with 30 sensors represents low coverage, while 200 sensors corresponds to high coverage. The model trained using all available simulation data and evaluated on its ability to reconstruct the corresponding fields under varying sensor placements. Besides, due to the computational cost of DiffusionPDE and S ""GM-cachh reconstruction involving thousands of model evaluations we restrict their evaluation 500 randomly sampled cases.
Results As shown in Table 2 PhySense achieves state-of-thh-art performance among all sensor counts, surpassing the second-best method by substantial margin. Despite starting from strong

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Table 2: Performance comparison on the turbulent flow and sea surface temperature benchmarks, Rel- ative L2 eтror is reported. PhySense-ops denotes the performance under optimized sensor placement, while all other deep models, including PhySense, use randomly placed sensors. Besides, SSPOR is classical method that performs both placement optimization and reconstruction. # indicates the sensor number. Promotion refers to the relative error reduction of PhySense-opt w.r.t. the best baseline.
Placement
Turbulent Flow
Sea Temperature
Models
Strategy
#200
#100
#50
#30
#100
#50
#25
#15
SSPOR
SSPOR
0.3018
0.4429
0.5921
0.6637
0.0756
0.0752
0.0789
0.0719
VoronoiCNN 14
0,3588
0.4870
0.6919
0.7992
0,1496
0.1537
0.2433
0.2817
Senseiver 41
0.1842
0.2316
0.3740
0.5746
0,0715
0.0732
0,0769
0.0784
GM 28
Random 0.1856
0.2016
0.2681
0.6421
0.1115
0.1137
0.1581
0.1795
DiffusionPDE:
18
0.1993
0.3312
0.6604
0.9163
0.0664
0.0775
0.0859
0.1139
PhySense
Random
0,1233
0.1527
0.2586
0.5176
0.0439
0.0452
0.0477
0.0520
PhySense-opt
Optimized
0.1106
0.1257
0.1558
0.2157
0.0426
0.0430
0.0437
0.0439
Relative Promotion
39.96%
38.56%
41 41.89%
62.46%
35.84%
41.26%5
43.17%
38.94%
Reconstruction
numbers
study: Turbulent Flow
Resonnttrattoo Emor
Reconstruction Error
PyySense-randko PhySense SSPOR
Myyeesse-pp
1E1711011
Sensor Numbers
Optimized Sensor Placement SSPOR Selected Placement
Random Sensor Placement
Figure 4: (a) Reconstruction loss versus the number of sensors. Random sampling, classical SSPOR, and our optimized method are compared using the same PhySense reconstruction base model. (b) Visualization of reconstruction results, error maps, and sensor placements (denoted by *). Our optimized placement clearly outperforms the other two strategies on turbulent flow benchmark.
reconstruction model, the reconstruction accuracy further significantly improved through sensor placement optimization. The improvement is particularly notable under low coverage with 30 sensors relative to the high coverage setup with 200 sensors. Remarkably, even with fewer sensors, the optimized 30-sensor placement exceeds the performance of the unoptimized 50-sensor counterpart.
As shown in Fig. 4 our learned placement strategy consistently yields the lowest reconstruction eгror compared to the other two strategies, especially in low coverage settings, demonstrating its superior effectiveness. Fig. bb) further supports this observation: without any explicit spatial constraints, the learned sensors are distributed in way that avoids excessive concentration, thereby reducing redundancy and enhancing information gain under the same number of sensors.
4.2 Sea Temperature
This benchmark is derived from the GLORYS12 reanalysis dataset 21]. focusing on the sea surface temperature variable. The data are downsampled to a 1 1 resolution, resulting in global ocean fields of size 360 180. During training, between 10 and 100 sensors randomly selected, with all sensors uniformly sampled from valid ocean regions, excluding land areas. The reconstruction task aims recover dense temperature fields from these sparse observations. Our training set comprises 9,843 daily samples from 1993 to 2019, with data from 2020-2021 reserved for testing. This setup allows us evaluate two critical aspects of model capability: (1) adaptability to varying sensor placements, and (2) generalization to temporal distribution shifts. Morcover, the presence of complex land-sea boundaries introduces significant challenges for both reconstruction and sensor placement optimization. We address this by providing land mask as an additional input.
Setups
PhySense-opt consistently achieves state-of-the-art performance across all sensor numbers (average improvement 40.0%), as shown Table Besides, SSPOR [32 performs reasonably well on this benchmark, due to the relatively stable temporal variations of sea temperature over the 30-year
Results
7

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Reconstruution error sensor numbers
Case study: Sea Temperature
Reconsttuution Enror
PhySense-randan PhySense-SSPOL
Reconstruction Enror
PhySense-opt
Sensor Numbers
Optimized Sensor Placement SSPOR Selected Placement Random Sensor Placement
Figure 5: Different placement strategies comparison and case study the sea temperature benchmark.
period. This also explains its poor performance on the turbulent flow, highlighting the limitations of traditional methods capturing complex, non-stationary physical fields.
The optimized sensor placement improves performance by an additional 12.0% over the unoptimized counterpart in the low coverage setting (25 and 15 sensors). As shown in Fig. a). the placement selected by SSPOR performs comparably to random selection, underscoring the importance of jointly co-optimizing reconstruction modeling and sensor placement optimization. Remarkably, just 15 optimally placed sensors covering only 0.023% of the spatial domain are sufficient to achieve sensors-covering reconstruction accuracy comparable with the best settings on this benchmark. which highlights the feasibility of global ocean temperature field reconstruction with limited sensing resources.
Moreover, our placement strategy successfully avoids land regions and performs effective optimization under intricate land-sea boundaries (Fig. 5]b)). Such counterintuitive optimized placement is difficult obtain through musan-designed heuristics. The error map within the red box clearly indicates optimized placement captures more information, which in turn improves the reconstruction accuracy.
4.3 Car Aerodynamics
Setups This benchmark is the first to address physies sensing tasks on 3D irregular meshes, We select fine 3D car model with 95,428 mesh points and simulate its pressure field using OpenFOAM (see Appendix [.T for simulation details). The sim- ulations span driving velocities from 20 to 40 m/s and yaw an- gles from -10 to 10 degrees. We simulate 100 cases in total, split- ting them into 75 for training and 25 for testing. The task involves reconstructing the surface pressure field from 10 to 200 randomly sampled sensors placed exclusively on the vehicle surface, since placing sensors in the surrounding but off-surface regions presents prac- tical challenges. This setup imposes particularly challenging conditions for placement optimization due to the complexity of the 3D surface and its geometric constraints.
Table 3: Main results on the car aerodynamics benchmark. Due resource constraints, only the best deterministic baseline and generative baseline are selected. To support irregular meshes, the backbone of DiffusionPDE [18) is replaced with Transolver [48).
Car Aerodynamics
Models
#200
#100
#50
#15
Senseiver
41
0.1009
0.1012
0.1018
0.1022
0.1053
DiffusionPDE
18
0.0967
0.0966
0.0987
0. 0.1555
0.2095
PhySense
0.0375
0.0382
0.0395
0.0416
0.0465
PhySense-opt
0.0369
0.0370
0.0370
0.0372
0.0386
Promotion
61.84%
61.70%
62.51%
63.60%
63.34%
Results As shown in Table PhySense-opt consistently achieves state-of-the-art performance across all sensor counts (average improvement 62.6%) Moreover, our placement strategy proves effective, especially in low-coeerage scenarios. With only 15 optimized sensors, we achieve a 17.0% improvement (from 0.0465 to 0.0386), and the reconstruction accuracy matches that of using 100 randomly placed sensors. Moreover, Fig, Tillustrates that, since the wind approaches from the front of the driving vehicle, regions such as the front end and side mirrors experience significant pressure gradients, likely caused by flow separation or viscous effects. Our optimized sensor placement successfully captures these high-variation regions, validating the effectiveness of the projected gradient descent optimization. Furthermore, our optimized sensor placement can be deployed in real-world settings, such as wind tunnel experiments, to collect actual pressure data. This enables correction of discrepancies simulation results and even of underlying physical systems, reducing the requirements for repeating costly real experiments and narrowing the sim-to-real gap.

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Cross Attention Visuaiization
b Intial Placement Sensitivity
Efficiency Analysis
Sanser postion
ffusionPDE
Eiroe (Jog
53GM
PhySense
Intall placament
Optimbeed placament Falativs 12 0.0335
033
506
200
02 0.8
Intial placement Optimbzed placemant Relative L2 0.0336
Ganeration Step Numbers 0og soalo)
Figure 6: (a) The cross-attention map visualizations from the last layer of PhySense, where sensor positions are pointed by red arrows. (b) Initial placement sensitivity analysis shows that varying initial sensor placements converge to different but equally effective placements, achieving similar reconstruction accuracy. (c) Efficiency comparison turbulent flow: these diffusion-based methods require 1,000 generation steps (100x slower than PhySense), but achieve even worse accuracy.
4.4 Model Analysis
Beyond the main results, we explore how to best incorporate sparse sensor information. We compare our incorporation strategy with an alter- native that concatenates an interpolated field-similar the approaches used in VoroniCNN [14). As shown in Table 4 this replacement leads to significant per- formance drop. especially under low spatial coverage (e.g., 30 sensors), highlighting the advantage of our method in capturing critical information from scattered sensor observations.
Ablations
Table 4: Ablation on sparse information incor- poration via an interpolated field (Rep. Inter.).
Turbulent Flow
Ablations
#200
#100
#50
#30
Rep. Inter. Ours
0.1571
0.2006
0.3584
0.7509
0.1233
0.1527
0.2586
0,5176
Cross attention visualization To demonstrate how PhySense dynamically associates sensor obser- vations with their relevant spatial influence regions, we visualize the cross-attention maps from the last cross attention block. As shown in Fig. ((a), PhySense captures physically meaningful regions of influence for each sensor on the complex mesh, indicating the underlying physical structure for its figh-fidelity reconstruction. Moreover, highlighted pair in the second row shows two spatially adjacent sensors exhibit diametrically opposed attention palterns due to local geometric variations.
Efficiency analysis The efficiency comparison on turbulent flow benchmark in Fig. O(c) shows that our method achieves significantly lower reconstruction errors across all step counts. Notably, while DiffusionPDE and GM require over 50 steps to achieve the error below 0.5, our method consistently maintains relative L2 error below 0.2 even with as few as steps. This efficiency advantage stems from fundamental differences in sparse information incorporation strategies. DiffusionPDD and S S'MM indirectly modify the data generation gradient based on sparse observations which require numerous iterative refinements to achieve proper guidance, while our approach directly conditions the model adaptively on sparse inputs during training. This architectural difference enables immediate and effective utilization of observational information from the very first inference step. As result, we achieve better performance with only I 1% model evaluations required by these baseline methods
Initial placement sensitivity During the sensor placement stage, different placement initializations lead to varying final sensor distributions. Meanwhile, as shown in Fig. G(b), after placement optimiza- tion, the reconstruction performance remains consistently high-quality and comparable across these variants, indicating that our placement optimization process not sensitive different initializations.
Progressive sensor placement optimization key advantage of our flow-based reconstructor is its in- herent support for progressive sensor placement op- timization. Unlike non- seneeeerraaive models, which operate on static inputs (e.g., pure noise or zeros), our model processes a noise-data mixture X, throughout the flow. This design inherently embeds a spectrum of reconstruction difficulties, corresponding to differ- ent noise levels, thereby amortizing the optimization process across the entire flow. Consequently, it provides more informative, tractable feedback for
Table 5: Ablation on progressive sensor place- ment optimization via pure noise input.
Turbulent Flow
Ablations
#200
#100
#50
#30
noise-inut mix-input (origin) Promotion
00.1118
0.1293 0.1630
0.2286
0.1106
0.1257
0.1558 0.2157
1.07% 2.78% 4.42% 5.65%

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Table 6: Performance comparison with four additional placement baselines on the turbulent flow benchmark, measured by relative L2 enror. Results obtained with limited sensors (e.g., 30) are highlighted in blue and the top-performing methods with comparable results are shown in bold.
Senseiver
PhySense
Placement Strategies
#200
#100
#50
#30
#200
#100
#50
#30
Random Sample
0.1842
0.2316
0.3740
0.5746
0.1233
0.1527
0.2586
0.5176
Grid Sample
0,1615
0.2273
0,3845
0,5869
0.1235
0.1521
0.2436
0.4930
Min-Max Sample
0.1549
0.2091
0.3795
0.5747
0.1196
0.1533
0.3157
0.5022
Our PGD on reconstruction model
0,1388
0.1750
0.3025
0.4570
0.1106
0.1257
0.1558
0.2157
Optimized Sensors from PhySense
0.1533
0.1833
0.3035
0.4601
0.1106
0.1257
0.1558
0.2157
sensor placement. An ablation study shown in Table 5 confirms that replacing the noise-data mix input with pure noise input in the feedback loop leads notable performance degradation, which is more pronounced with fewer sensors, underscoring the unique benefit of this flow-process feedback.
Placement generalization We compare our placement strategy with four more placement baselines across Senseiver and PhySense under different sensor numbers turbulent flow dataset.
(i) Grid: We first place sensors uniform grid using the largest square number within the budget, and then randomly allocate the remainder. This strategy guarantees spatial coverage but remains ag- nostic to the physical dynamics. Practically, its performance is similar to random placement, aligning with classical sampling theory [4] that questions the efficacy of grid sampling for complex systems.
A heuristic method that promotes spread-out sensor placements by iteratively placing the next sensor at the point that maximizes the minimum distance to existing sensors. However, this approach does not yield significant improvements over random placement in our experiments.
(ii)
(iii) Our Projected Gradient Descent: An upper-bound method using any reconstruction model (Sen- seiver or PhySense) and performs our projected gradient descent to optimize sensor placements.
(iv) Optimized Sensor from PhySense: To evaluate the generality of the learned placements by Phy- Sense, we feed the placement into the Senseiver. The performance closely matches that of Our PGD and significantly outperforms all other baselines, particularly when the number of sensors is limited (e.g.. 30), challenging scenario to place sensors. This suggests that our learned placements effectively capture physically informative structures that generalize beyond specific reconstructors.
Optimized sensor distribution visualization To explore the distribution of optimized sensors, we visualize their relationship with information distribution (measured by the model's spatial gradient magnitude) and variance distribution (point-wise vari- ance of the physical field) on turbulent flow benchmark. Inter- estingly, most of the optimized sensors lie within nigh-variance regions. More importantly, each sensor is located near a high- information point, and their spatial distributions exhibit similar patterns. This suggests that the optimizer is not merely clustering high-variance areas but strategically identifying spatially sep- arated, high-sensitivity locations which capture the field's global dynamics. This aligns with the sensing principle that maximal information gain requires sampling diverse, non-redundant points rather than densely covering localized variance hotspots.
Information Distribution
Variance Distribution
Figure 7: Visualization of infor- mation and variance distribution on turbulent flow benchmark.
5 Conclusion
To co-optimize physies field reconstruction and sensor placement for accurate physics sensing, this paper presents PhySense as synergistic two-stage framework. The first stage develops base model through flow matching that reconstructs dense physical fields from all feasible sensor placements. The second stage strategically optimizes sensor positions through projected gradient descent to respect geometry constraints. Furthermore, our theoretical analysis reveals a deep connection between the framework's is objectives and the classical A-optimal principle, i.e., variance minimization. Extensive experiments demonstrate state-of-the- performance across benchmarks, particularly on a complex 3D irregular dataset, while discovering informative sensor placements previously unconsidered.
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Acknowledgments and Disclosure of Funding
This work was supported by the National Natural Science Foundation of China (62021002 and U2342217), the BNRist Innovation Fund (BNR2024RC01010), State Grid Ningxia Electric Power Co. Science and Technology Project (SGNXYX00SCIS240003)) and the National Engineering Research Center for Big Data Software.
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16. Declaration of LLM usage
19

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Question: Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods this research? Note that if the LLM used only for writing, editing, or formatting purposes and does not impact the core methodology. scientific rigorousness, or originality of the research, declaration is not required.
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Please refer to our LLM policy for what should or should be described.
20

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A Proofs of Theorems in the Method Section
A.1 Proof of Theorem
Proof. By the pointwise optimality of the squared loss in the training flow loss ①
px pxx, Xo
X, x, i, P. (P))
Taking the conditional expectation of X1 gives
[[X1 p, (p) = E Xo + S v, dt p, x((b)
E|x p. XI(D) E (x
p, Xi(p)
(A)
(B)
Step Move the expectation inside the integral (Fubini's theorem):
(B) ( E ['
p. X1(p) dt.
Step 2: Substitute the definition of v* and apply the law of total expectation:
E (xx.......))) p. {1(P)
E E E Xo X, p. x1(p)
E X - Xo p. (P))
The right-hand side no longer depends on t, hence
(B) = J E[x, Xn p. (p) = E(x, - Xn p. x (p)].
Step 3: Combine (A) and (B):
(A) + (B) E X, p. (1 + Xx p.p (p) E p. X.
Therefore, flow-based reconstructor is an unbiased estimator physial fields.
A.2 Proof of Theorem
3.5
Part 1. First, we will prove LA (p) = - = Ed Ap(p) with eigenvalues {A<()) of Ep.
Because X, (p.))(p)) NO Ep). we have
(P E X EX
[|]]
tu Var(X))) tu Ep)
Let the orthogonal decomposition of the placement-dependemt covariance matrix be
Σp = UAUT A
Aa(p))
where U is an orthogonal matrix, i.e. UUT = J. where U is an orthogonal matrix, i.e. UUT = I. All eigenvalues (p) are non-negative because the covariance matrix Σ, is positive semidefinite. Since the trace is invariant under orthogonal similarity transformations, we obtain
(x(P) t((pp) = ir(A) Σ (P)
(A.0)
21

---

Part 2. Second, we will show,
Lfow (p)
B8 [|(x - Xo) - -x de
n
un Var((x - Xo x,,IP..((P)))
(p(p),
Step 1: Calculate the conditional variance Van(X1 Xo X1(p))
We denote that,
X Z =
Σz di)g(T,Ep), Yx - Xo, X, (1=11111) -)x, :X1 (0 Since X, and X, are independent, Z conditioned on (p. (p)) remains Gaussian,
X1
z (p. X1(p)) ~ A(0 Ez)
Writing Y and X, as linear maps of Z,
Y VZ,
V (-(-1, I).
= C= ((1 t)1, tl),
thus the pair (Y, X,) is jointly Gaussian. The standard conditional-cooariance formula yields
Var Y (x VE2V" - (CEC"( CEZ1
(A.1)
with
VEzV = -1+ Epp
CΣNC (1 t)21 2
VE2C = (1 - (1-1)] 1Ep
Substituting into AA.l
gives
Var(Y X,, (. (1-0 -1) (1p--(11p))) (1Ep (1-t) (A.2)
Step Diagonalisation and scalar reduction
(1()) and denote it by T.p)). Because Σp UAUT for orthogonal matrix U and diagonal matrix A
Recall from (.2)) the matrix form of the conditional variance Var Y t,
Aa), all terms A.2 commute with U: consequently
[x(Ep) = UT.(A)UT.
Thus,
tr(T((Ep) = Lr(T.
Consequently, the flow-male now-matching objective can be written entirely in terms of the diagonal matrix A:
Lnow (p) S U[Van(x1 Xxo X,,t,,, de
(r(T,(E.))) di
(r(T((A)) de
/ r((x(p)) dt.
Σ
2m
22

---

where the last equality uses that Tz(A) = (AA))..(A))......... ..... P((Au)) ((((a)))..,((()))) is diagonal, so its trace equals the sum of its diagonal entries.
According to the expression of rt
Lfow (p) = Σ/
x i=1 E22 + (1 ()- dt, A 20
(A.3)
Step 3: Calculate the integral above over 1 ~ 11, 1|.
For À > 0 and consider
I(A) = Jt 1*+ (1- dt.
Set A := X+ 1. Completing the square in the denominator gives
xf2 + (1- (A (A 28 1==
2) + A
First substitution. We introduce
A (1-
A = V A
u
VX
du,
so that
1
uo
f=1
U1
VAA
A
Substituting, obtain
X u1
du
21
du
T(A) =
VAA
A
u + A-1
Au +
Second substitution. Absorb the constant A-1 by setting
dv
v VAu,
du
VA
which converts the integrand to the standard form (u² + 11-1 The limits become
Up V
U1 VAm==V
Collecting the Jacobian factors yields
V
du
dv
(()) VAA
VT
2 +1
-1/VX 2² +1
The integrand is elementary, giving
I(A) vā Jarctan u] ==
x= 1/VX
VA (arrtan VA arctan
Vā arrtan Vā + arctan )x)
For any x > 0 the identity arctan + arctan ! = T/2 holds. Then substituting V yields,
I()) VA.
For À = 0, the identity also holds obviously.
23

---

Applying this to every X; in A.3 gives
Lrow (p) Σ 2 V >p((p)
which is the target of Part 2.
Part We prove 3/2 flow (p) (P(P) S Roo (p).
By the cechmeetic-gemeetrii mean inequality, we have,
Σ A 2m
(p) S 2 2
=
SLA 5
Σ mm
2
im1
Therefore,
L (p) S (A <
Ciow (p)
which completes Part and hence the proof of Theorem 3.5
Remark A.1. Furthermore, under Assumption 3.2 this mutual control relationship naturally extends both D-optimality and E-optimality objectives. Specifically, their formulations simplify
Cp(p) = dot(pp) -II Xp)). CE(P) -mmax (Ep).
It evident that both LD and LE be polynomially bounded above and below by LA due to standard spectral inequalities. Consequently, through the control of L. the flow-loss objective also polynomially controls CD and Lg in both directions. Therefore, the flow-loss objective serves as spectrum-awee aware surrogate that can also guide D-optimal and E-optimal sensor placement with bounded distortion.
B
Full Results of Sensor Placement Optimization
Due to the space limitation of the main text, present the full results of sensor placement results turbulent flow and sea temperature here, as a supplement to Fig. 4 and Fig.
Given the same high-capacity reconstruction model, our optimized placement consistently achieves superior performance compared SSPOR and random baselines, confirming the effectiveness of our joint strategy. Interestingly, SSPOR does not always outperform random sampling: for example, on the sea temperature dataset, it often performs slightly worse. This underscores the importance of co-optimizing sensor placement alongside the reconstruction process.
Table 7: Performance under different placement strategies turbulent flow and temperature.
Placement Strategy
Turbulent Flow
Sea Temperature
Models
#200
#100
#50
#30
#100
#50
#25
#15
PhySense
Random
0.1233
0.1527
0.2586
0.5176
0.0439
0.0452
0.0477
0.0520
PhySense
SSPOR
0.1143
0.1348
0.1815
0.3397
0.0452
0.0463
0.0454
0.0543
PhySense
Optimized
0.1106
0.1257
0.1558
0.2157
0.0426
0.0430
0.0437
0.0439
Relative Promotion
3.23%
6.75%
14.16%
36.52%
2.96%
4.87%
3.74%
15.58%
C Implementation Details
In this section, we provide the implementation details of our experiments, including benchmarks, metrics, and implementations. All the experiments are conducted based PyTorch 2.1.0 34 and on A100 GPU server with 144 CPU cores.

---

C.1 Benchmarks
While the turbulent flow and sea temperature benchmarks are described in detail the main text, here we focus on elaborating the key details of our generated car aerodynamic benchmark.
a Computation domain
b Slice of the refined mesh
Figure 8: (a) Computational Domain of the CFD simulation. (b) Refinement regions around the surface.
We watertight 3D mesh taken from the ShapeNet VI 6 dataset, which contains around 50k triangular facets. The aerodynamic pressure of the car surface simulated using OpenFOAM [20]. As shown in Fig- [[)), create virtual wind tunnel to simulate the airflow around width, length and height of the wind tunnel are set to 16 meters, 30 meters and 8 meters respectively. The computational mesh around the car surface is refined to a smaller mesh size for more accurate acrodynamics modeling, which is illustrated in Fig. b)..
The
We specify different inlet velocities and yaw angles for different simulation cases. The velocity ranges from 20 m/s to 40 m/s, while the yaw angles range from -10 degrees to 10 degrees. We utilize Reynolds-Averee
Simulation (RAS) with the k-omega SST turbulence model in OpenFOAM. 600 simulation iterations are conducted for each case to ensure convergence. A no-slip boundary condition of the velocity field is defined the surface. We simulate 100 cases in total, splitting them into 75 for training and 25 for testing.
C.2 Metrics
To evaluate the accuracy of the reconstructed physical field, employ the widely used relative L2 metric, defined as
||
Relative L2
u
where u denotes the reference (ground-truth) field and represents the model reconstruction.
C.3 Implementations
We consider the following baselines on the turbulent flow and sea temperature benchmarks:
Reconstruction models: SSPOR, VoronoiCNN, Senseiver, S*GM, and DiffusionPDE.
Sensor placement strategies: random sampling and the SSPOR's 's sensor selection method.
Due to the geometric complexity the car's 3D surface, we compare against the top two performing reconstruction baselines Senseiver and DiffusionPDE-ffr for this task. The common setups for all models are shown the Table
g
Table 8: Common setups for all models in three benchmarks. Other setups follow their original setup.
Benchmarks
Turbulent Flow
Sea Temperature
Car Aerodynamics
Epochs
800
300
300
Batch Size
60
40
Learning Rate
10-3
2 10
10
Optimizer
ADAM
We utilize the official implementation of SSPOR, the PySensor package to evaluate the reconstruction and sensor selection performance of SSPOR. SSPOR employs POD to express
SSPOR 32
25

---

high-dinme dith-simensii data as a linear combination of several orthonormal eigenmodes obtained from the singular value decomposition (SVD). Specifically, we employ the pysensors. SSPOR class with an SVD basis to obtain the targeted number of sensors from the training data. The selected sensors are then used to reconstruct dense fields from sparse observations either with SSPOR or our method.
VoronoiCNN 14
VoronoiCNN integrates Voronoi tessellation with convolutional neural networks (CNNs). However, the official implementation uses simplistic CNN that struggles with complex and dynamic fields. We replace it with a more powerful U-Net [38] architecture, using 64 base feature channels, with downsampling and upsampling layers and skip connections to enhance information flow. Due to its inability to handle irregular car aerodynamic meshes, experiments are conducted only on turbulent flow and sea temperature datasets under a unified setup.
Senseiver 41
We utilize the official implementation of Senseiver. Senseiver is built upon an implicit neural representation (INR) architecture, which optimizes reconstructions at individual spatial locations rather than over the entire physical field. Specifically, during training, 2048 points are randomly sampled from each field to supervise the model. This inherent sampling randomness further impacts the stability of Senseiver and contributes to its slower convergence. To better assess its full potential, we did not strictly enforce the same training epoch limit used for other baselines. Instead, we allowed Senseiver to train until the default early stopping criterion in its official implementation was triggered. Across the three distinct benchmarks we use, the hyperparameters of Senseiver architecture are carefully tuned based on the official implementation scripts.
In our experiments, we follow the official implementation and conduct training and inference. Its backbone the full U-Net architecture equipped with attention mechanisms and timestep embeddings, We find that when the number of iterative optimization steps is less than 100, S'GM fails to produce clear generation results. Moreover, the generation results can be significantly affected by the random sampling strategy of the sensors, with some cases exhibiting abnormally high relative errors, even greater than 10. Therefore, we increase the number of iterative optimization steps to 1,000 obtain comparable experimental results.
GM 28
DiffusionPDD [18]
We borrow the model architectures of the original implementation of Diffu- sionPDE for the turbulent flow and sea temperature benchmarks, and employ a Transolver [48 as the backbone for the unstructured car aerodynamics benchmark. For inference, we also follow the original implementation configurations, using total of 2,000 steps for better performance. The guidance weights we search each benchmark are shown in Table 9
Table 9: Guidance weight of the observation loss for DiffusionPDE. Our final selection is bolded
Benchmarks
Turbulent Flow
Sca Temperature
Car Acrodynamics
Guidance Weight
{1000, 2500,
[1000, 2500, 5000)
( (1000, 2500,, 2500,
PhySense For the reconstruction model, we use DiT as base models for turbulent flow and sea temperature benchmarks, and switch Transolver for car aerodynamics benchmark. The detailed model configurations are summarized in Table I0
Table 10: PhySense model configurations for different benchmarks. "_"" in the patch size column indicates that Transolver does not apply the patchify operation.
Benchmark Patch Size
dim
Depth
Heads
dim head
mlp_dim dim
Others
Turbulent Flow
(2, 2)
374
8
32
374
Sea Temperature
(3, 3)
374
12
32
374
Car Acrodynamics
374
12
8
32
374
slice_num=32
For sensor placement optimization, we adopt the Adam optimizer to perform gradient-based updates, followed by a projection step implemented via nearest-neighbor search to satisfy the spatial constraints. All models are trained for 5 epochs with a cosine learning rate decay scheduler. The learning rate closely tied to the spatial scale of the physical domain and carefully tuned for each setting, as summarized in Table [I]
26

---

Table 11: Learning rate sensor placement optimization on three benchmarks.
Benchmarks
Turbulent Flow
Sea Temperature
Car Aerodynamics
Learning rate
0.25
0.0025
D
Additional Analysis
D.1
Comparsion with AdaLN-Zero
Here, compare our sparse sensor incorporation strategy turbulent flow with widely used conditioning approach in diffusion models, namely the AdaLN-Zero block in DiT [35]. As shown in Table [2 replacing design with AdaLN-Zert leads substantial degradation in performance, with reconstruction errors approaching 1. This is because sensor placement inherently captures localized information, whereas Adal.N-Zee aggregates all sensor data into single global variable, which is insufficient for preserving the spatial specificity required in reconstruction tasks.
Table 12: Ablation on sparse information incorporation via A- block, i.e., Rep. N-Zer-
Sensor Number
#200
#100
#50
#30
Rep. AdaL--Zero Ours
0.9179
0.9486
0.9695
>1
0.1233
0.1527
0.2586
0.5176
D.2 Hyperparameter Sensitivity
As adopt projected gradient descent optimize sensor placement, the learning rate in Eq. 2 becomes a critical factor, especially when operating under complex geometric and high-dimensional surface constraints. We investigate the learning rate sensitivity on the aerodynamics benchmark. As shown Table the optimization process relatively insensitive to learning rate variations when the number of sensors exceeds 30. In contrast, the 15-sensor the spatial domain exhibits noticeable sensitivity, indicating the increased difficulty of optimization under extreme sparsity. We ultimately select 17 0.0025 the final choice.
only 0.016% of
Table 13: Impact of learning rate sensor placement optimization in the car aerodynamics bench- mark. We ultimately select 7) = 0.0025 as the final choice, which is bolded.
Learning rate
0.005
0.0025
0.001
0.0005
30 sensors 15 sensors
0.0379
0,0372
0.0369
0.0372
0.0423
0.0386
0.0389
0.0400
D.3
Statistical Analysis
Table 14: Evaluation results of the PhySense reconstruction model under random sensor placements with varying sensor counts (10-200) on car acrodynamics benchmark. For each setting, the recon- struction repeated three times, and the mean and variance of the relative L2 loss reported.
Sensor Number
#200
#100
#50
#30
#15
Sampling
0.0376
0.0386
0.0396
0.0414
0.0458
Sampling 2
0.0379
0.0378
0.0387
0.0414
0.0436
Sampling 3
0.0370
0.0382
0.0401
0.0419
0.0501
Mean Standard derivation
0.0375
0.0382
0.0395
0.0416
0.0465
0.0004
0.0003
0.0006
0.0002
0.0027
In this section, we analyze the statistical significance of our experimental results. Our pipeline consists of two stages: training the reconstruction model and optimizing the sensor placement. Due to the high computational cost of training the base reconstruction model, we train it only once. The sensitivity of sensor placement initialization has been discussed in Section 4.4 Our findings show that while different initializations result in different placements, each optimized placement is effective.
27

---

Table 13: Evaluation results of the PhySense reconstruction model under optimized sensor place- ments with varying sensor counts (10-200) on car aerodynamics benchmark. For each setting, the reconstruction is repeated three times, and the mean and variance of the relative L2 loss are reported.
Sensor Number
#200
#100
#50
#30
#15
Sampling 1
0.0369
0.0371
0.0369
0.0370
0.0378
Sampling 2
0.0367
0.0372
0.0369
0,0374
0.0389
Sampling 3
0.0370
0.0368
0.0373
0.0373
0.0390
Mean
0.0369
0.0370
0.0370
0.0372
0.0386
Standard derivation
0.0001
0.0002
0.0002
0.0002
0.0005
Furthermore, since task is reconstruction problem, and our reconstruction model essentially learns a conditional distribution, we report in the table above the mean and standard deviation of reconstruction loss under multiple samplings on the car aerodynamics benchmark, across varying numbers of sensors. As shown in Table [4 and [5] when the number of sensors exceeds 15, the standard deviation becomes relatively small, indicating stable performance. Moreover, regardless of the sensor count, once the placement is optimized, the reconstruction results exhibit consistent stability across multiple runs.
D.4 Additional Metric for Gradients
Table 16: Comparison on sea temperature benchmark. Relative L2 error gradient fields is reported.
Sea Temperature (relative L2 of gradient field)|
#100
#50
#25
#15
Senseiver + random sensor placement
0.5456
0.5470
0.5501
0.5515
PhySense random sensor placement
0.4877
0.4884
0.4890
0.4892
Relative promotion
11.87%
11.98%
12.51%
12.74%
In this section, we additionally evaluate the relative L2 error of the gradient fields on the sea temperature better reflect whether the model captures meaningful physical fidelity. As shown the Table [6] PhySense achieves significantly lower gradient error compared to the second-best model, Senseiver, demonstrating its superior spatial physical fidelity.
D.5
Uncertainty quantification
Sample
Sample
Sample
Sample Mean
Sample Variance
Erar of Sample Moan
Figure 9: Visualization of uncertainty quantification for the flow-based reconstruction model on the turbulent flow benchmark. The sample variance is notably smaller than the physical field and error.
Flow-based reconstruction models explicitly learn the full conditional data distribution, which inherently facilitates uncertainty quantification. complete this, we reconstruct each input three times, each with distinct random noise initialization. Subsequently, we compute the pixel-wise sample mean and sample variance across these reconstructions. illustrated Fig. g the magnitude of the variance is orders of mugnitude smaller than that of the sample mean and the reconstruction error. This indicates high confidence in our model, consistently produces similar reconstructions irrespective of the initial noise vector.
28

---

E
Limitations and Future Work
While our current formulation assumes point-wise sensor observations, this may not fully capture the behavior of certain real-world sensors that integrate information over small spatial neighborhoods. For example, radar and precipitation sensors typically respond local regions rather than single points. Incorporating regional observation into both the reconstruction model and the sensor placement optimization could further improve effectiveness of our approach. As a future direction, we aim to extend both our model and theoretical framework from point-wise observations to region-based sensing, which better reflects the behavior of many real-world sensors.
F Boarder Impacts
We introduce PhySense, synergistic two-stage framework for accurate physics sensing with theorec- tical guarantees. Extensive experiments validate its superior reconstruction accuracy and informative sensor placements. Beyond performance gains, work offers new perspective the sensor placement problem from deep learning standpoint. Thus, PhySense may inspire some future research in relevant domains. Moreover, optimized placement could potentially inform how sensors are placed in more applications, enhancing their reconstruction accuracy.
Since our work is purely focused algorithmic design for accurate physics sensing, there are no potential negative social impacts or ethical risks.
29


Reviews:
[{"review_id": "0bQKo2PHA8", "reviewer": "Reviewer_LEF1", "summary": "PhySense proposes a two-stage reconstruction model that uses an efficient flow based generative model to infer a reconstructed physical field conditioned on a set of sparse pointwise sensor measurements, then updates sensor positions by performing a gradient step that improves reconstruction accuracy and then projects sensor positions back to the domain (e.g. any set the sensors are restricted to). This process is repeated for a fixed number of rounds and –as the paper proves–is guaranteed to produce the A-optimal (e.g. variance minimizing) set of sensor positions. The novel use of flow matching in this setting already leads to improved one-shot reconstructions of physical fields over a range of prior works and the additional optimized sensor placement further improves performance. The framework is general enough that–as the paper demonstrates in experiments–the particular architecture of the velocity field can be interchanged depending on the specific application.", "questions": "* Does PhysSense take additional measurements at each step of projected gradient descent as its optimizing sensor positions? It's not clear to me whether the sensor optimization is a performed with real measurements or generate data.\n* Where does this method stand in relation to approaches like Thompson sampling? Specifically, I'm curious if PhySense can be applied to adaptive sampling problems, where \"sensors\" are not so much a physical device that is being positioned but rather a position at which a measurement is collected (e.g. a camera moving around an object)?\n* On line 191 is the conditional random variable supposed to be centered on 0 or X1(p)?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "strengths:\n1. The presentation is exceedingly clear, providing excellent background and intuition on the three core areas leveraged in the method: sparse reconstruction, optimized sensor placement, and flow-based matching. I found the method section very easy to follow as a result, despite the fact that it incorporates many disparate subfields.\n\n2. The method is straightforward, yet relies on an elegant theory proving that A-optimality is obtained through performing gradient descent on the flow residual with respect to the sensor positions. This raises many interesting follow up questions around how to achieve alternative sensor placement objectives–some of which were summarized earlier in the paper–and the interplay these objectives have on reconstruction methods.\n\nweaknesses\n\n1. The comparison against other reconstruction methods is partially unclear to me. It seems like in the process of optimizing sensor position, PhysSense makes additional measurements and this leads to more effective sensor positions being utilized than in the naive case–even if an equal number of measurements are directly applied to reconstruction. This could also be a misunderstanding on my part though, either in how PhySense is utilizing its sample budget or whether the sensor optimization is run as a pre-processing step using only the domain geometry and sampled PDE solutions.", "was_revised": true, "final_justification": "The paper introduces a clear, well-motivated two-stage framework for reconstructing physical fields from sparse sensors and optimizing sensor placement, with theory guaranteeing optimality. My concern about whether optimization required new measurements was resolved in the rebuttal, which clarified it is done offline with historical data. With this, I view the work as technically solid, clearly written, and impactful, and maintain my strong acceptance recommendation.", "rebuttal": "We would like to sincerely thank Reviewer LEF1 for providing a detailed review with positive evaluations and insightful questions.\n\n>**Q1:** \"how to achieve alternative sensor placement objectives and  the interplay these objectives have on reconstruction methods\"\n\nIn our paper, we rigorously prove in $\\underline{\\text{Theorem 3.5}}$ that the flow-loss minimization objective polynomially controls the classical A-optimal criterion. Furthermore, under $\\underline{\\text{Assumption 3.2}}$, **this mutual control relationship naturally extends to both D-optimality and E-optimality objectives**. Specifically, their formulations can be simplified to \n\n$\\mathcal{L}\\_D$(p) := det($\\Sigma\\_p$) = $\\Pi_{i=1}^d$ $λ_i$(p)  \n$\\mathcal{L}_E$(p) := $λ\\_{max}$($\\Sigma\\_p$)\n\n\n\nIt is evident that both $\\mathcal{L}_D$ and $\\mathcal{L}_E$ can be polynomially bounded above and below by $\\mathcal{L}_A$ due to standard spectral inequalities. Consequently, through the control of $\\mathcal{L}_A$, the flow-loss objective also polynomially controls $\\mathcal{L}_D$ and $\\mathcal{L}_E$ in both directions. Therefore, the flow-loss objective serves as a spectrum-aware surrogate that can also guide D-optimal and E-optimal sensor placement with bounded distortion. \n\nWe will include this new finding in the main text and provide detailed proofs in the appendix to further demonstrate the generality and extensibility of our framework.\n\n>**Q2:** \"The comparison against other reconstruction methods is partially unclear to me.\"  \"How PhySense is utilizing its sample budget.\" \" whether the sensor optimization is performed with real measurements or generate data\"\n\nPhySense can be viewed as a pre-processing stage that leverages historical or pre-collected data. For instance, we utilize 27 years of global ocean reanalysis data with randomly sampled sensor placements to train a reconstruction model. This model then provides feedback to guide the optimization of sensor placement via projected gradient descent. In this way, PhySense jointly optimizes both the sensor placement and the reconstruction model with respect to the data distribution. **During actual deployment (inference time), only the optimized sensors are used to collect real measurements—thus, the sample budget strictly equals the number of deployed sensors.** These measurements are then passed through the trained reconstruction model to reconstruct the full physical field. \n\nJointly optimizing sensor placement and reconstruction is a key contribution of our work. **To isolate and evaluate the performance of the reconstruction models alone, we adopt a randomized sensor placement strategy as an idealized setting**. This allows us to assess reconstruction quality independent of the placement method. All baseline methods follow the same pipeline for a fair comparison across all tables when the placement strategy is set to random.\n\n> **Q3:** \"PhySense stand in relation to adaptive sampling approaches like Thompson sampling\"\n\nThank you for the insightful question. Although our current work focuses on static sensor placement, it is closely related in spirit to adaptive sampling strategies such as Thompson Sampling, which also aim to select informative locations under uncertainty. Given the importance and practical relevance of adaptive sensing, we view this as a promising direction for future work.\n\n> **Q4:** \"On line 191 is the conditional random variable supposed to be centered on 0 or X1(p)\"\n\nThe conditional Gaussian assumption in $\\underline{\\text{Assumption 3.2}}$ models the target distribution as centered at 0 **for notational simplicity**, which is a standard treatment especially when the **data has been normalized**. Indeed, in practice, the observed values at the sensor locations $\\mathbf{X}_1(\\mathbf{p})$ are fixed (i.e., conditioned on), and the conditional variance at $\\mathbf{X}_1(\\mathbf{p})$ is zero. Therefore, one could equivalently center the distribution around $\\mathbf{X}_1(\\mathbf{p})$ instead of 0. Furthermore, since **our theoretical analysis focuses solely on the covariance structure (i.e., the variance components), the specific form of the mean does not affect the results**. Thus, the derivations remain valid regardless of the centering chosen."}, {"review_id": "zuu0wQ7tNw", "reviewer": "Reviewer_X4JQ", "summary": "The paper proposes PhySense, a two-stage approach that both reconstructs a physical field from sparse sensing data, as well as optimizes the sensor locations to be optimal. The paper first uses a flow-based generative model to generate the physical field conditioning on a sparse set of measurement data. This learns a conditional generative model given input of different numbers and locations of the sensors. Then, the learned generator is used to refine the set of sensors' locations. The paper provides both theoretical and experimental guarantees about the convergence and performance gain compared to existing models.", "questions": "1. How is the flow loss calculated in Eq.(2)? I thought the optimization is performed during test time and the ground truth physical field (i.e., $X_1$) is not available. \n2. Assumption 3.2: Is this assumption reasonable? Why would it be reasonable to assume that the target physical field follows a zero-mean Gaussian distribution? \n3. The paper only evaluates the L2 loss of the modeled field. Is this a reasonable metric? I would imagine that the reconstructed physical field would be used for downstream applications that involve perhaps other quantities from the field (e.g., gradients, Hessians).", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n1. The paper has a strong theoretical section in which it proves some theoretical guarantees of their method's convergence. These theoretical analysis boosts the credibility of the paper's method and their experimental performance. \n2. Compared to previous works that only focus on reconstructing the physical field, the paper proposes a novel joint optimization of both the physical field and the sensor locations, showing that a joint optimization of both fronts would lead to better performance. \n3. The experimental section contains multiple settings covering a wide range of physical phenomena. The proposed method achieves non-trivial improvements over the existing methods in all these settings, demonstrating the effectiveness of their approach. \n\n## Weaknesses\n1. From reading the paper, it seems that no physical constraints are used when learning the generative model. Would adding some sort of physically informed loss would help with the reconstruction? \n2. I found it hard to motivate the physical meaning of the optimized sensor locations in the second stage of the method. The sensor location optimization seems to be finding the optimal sensor configuration based on the learned generative model, so I wonder if there is some kind of overfitting happening when doing the second stage optimization. For example, would the physical field reconstructed using other methods with the optimized sensor locations improve?", "was_revised": true, "final_justification": "After rebuttal, I am clear of my original questions and had a better understand of the paper. I believe the authors have done a thorough job in terms of theoretical grounding and experimentally validating their proposed joint sense + reconstruction pipeline. I think this joint learning approach enables a wide range of applications and future directions for research. Therefore, I raised my score to accept.", "rebuttal": "Many thanks to Reviewer X4JQ for providing a detailed review and insightful questions.\n\n> **Q1:** \"no physical constraints are used when learning the generative model. Would adding some sort of physically informed loss would help with the reconstruction\"\n\nFirst, we would like to emphasize that reconstruction is only one part of our framework—our primary contribution lies in the **joint optimization of sensor placement and reconstruction**, which is critical for accurate physics sensing.\n\nWhile we do not impose explicit physical constraints during model training, the data used in our setting already reflects underlying physical laws and thus **implicitly lies on a low-dimensional physical manifold**. This inductive structure, which is often absent in general vision datasets, is what makes the reconstruction process—from sparse to dense observations—both feasible and effective in physical domains.\n\nMoreover, in our target scenarios, we are **not aiming to reconstruct a fully closed physical system governed by complete equations** (common in real settings). For example, we focus on reconstructing the temperature of ocean surface rather than the full 3D ocean dynamics, or the surface flow over a car rather than the complete surrounding velocity field. **In such partially observed systems, it is often not feasible to impose full equation-based constraints**, as many physical variables are unobserved or unmeasurable. This limitation applies equally to all baseline methods we compare against.\n\nFurther, we agree that incorporating physically informed losses can be beneficial. While it is orthogonal to our current focus, our framework is complementary to such efforts and can potentially integrate with them in future work.\n\n> **Q2:** \"the physical meaning of the optimized sensor locations in the second stage of the method. The sensor location optimization seems to be finding the optimal sensor configuration based on the learned generative model, would the physical field reconstructed using other methods with the optimized sensor locations improve\"\n\nFirst, the optimized sensor placements learned in the second stage of our method exhibit **a certain degree of physical interpretability**. For example, as shown in Figure 5 on the sea temperature dataset, the selected sensors tend to concentrate near **coastal regions or ocean currents**, where the physical dynamics are most active and spatial variability is high.\n\nMoreover, we do not view the sensor optimization as overfitting to our reconstruction model. Rather, our framework **jointly learns**:\n\n1. The **global biases of the underlying physical field**, which are **model-agnostic** and thus transferable across reconstruction models.\n2. The **model-specific inductive biases**, which allow our flow model to extract more information from sparse observations.\n\nTo further investigate this, as per your request, we conducted an ablation study comparing:\n\n- **Senseiver + random sensor placement**\n- **Senseiver + sensor placement optimized by Senseiver using our PGD**\n- **Senseiver + sensor placement optimized by our PhySense**\n\nInterestingly, the third setting—Senseiver using sensor locations optimized by PhySense—yields **significantly better performance** than random placement, suggesting that our learned placement captures **physically informative structures** that go beyond the reconstruction model's own inductive bias. The quantitative results are provided in the following table.\n\n| Turbulent Flow                                               | 200 sensors | 100 sensors | 50 sensors | 30 sensors |\n| ------------------------------------------------------------ | ----------- | ----------- | ---------- | ---------- |\n| Senseiver + random sensor placement                          | 0.1842      | 0.2316      | 0.3740     | 0.5746     |\n| Senseiver + sensor placement optimized by Senseiver using our PGD | 0.1388      | 0.1750      | 0.3025     | 0.4570     |\n| Senseiver + sensor placement optimized by our PhySense       | 0.1533      | 0.1833      | 0.3035     | 0.4601     |\n| PhySense + optimized sensor placement                        | 0.1106      | 0.1257      | 0.1558     | 0.2157     |\n\n> **Q3:** \"How is the flow loss calculated in Eq.(2)? I thought the optimization is performed during test time and the ground truth physical field (i.e., ) is not available.\"\n\nThe optimization in Eq. (2) is not performed at test time. Rather, **it is part of the second stage of training, where ground-truth physical fields are fully available**. For instance, we utilize 27 years of global ocean reanalysis data with randomly sampled sensor placements to train a reconstruction model. This model then provides feedback to guide the optimization of sensor placement via projected gradient descent. **The two steps occur entirely during training time.**\n\nDuring actual deployment (inference time), only the optimized sensor locations are used to collect real measurements, which are then passed through the trained reconstruction model to recover the full field. **Performing sensor optimization per test case is impractical**, as it would require access to ground-truth fields unavailable in real world. Instead, our framework produces a **consistent, scene-specific sensor placement that generalizes across test cases and can be deployed in practice**.\n\n> **Q4:** \"Why would it be reasonable to assume that the target physical field follows a zero-mean Gaussian distribution\"\n\nThe assumption that the mean is zero in Assumption 3.2 is adopted primarily for **notational simplicity**, which is a standard practice especially when the data has been normalized. Importantly, our theoretical analysis focuses exclusively on the **covariance structure** of the field (i.e., the variance components), and the specific form of the mean does not influence the derivations or conclusions. As such, the results remain valid regardless of the centering.\n\nMoreover, the **Gaussian assumption is commonly used in theoretical formulations due to its analytical tractability**, particularly the ability to derive closed-form expressions for conditional distributions. This modeling choice has been widely adopted in prior works on sensor placement and experimental design [1].\n\nFurthermore, we emphasize that this assumption is made solely for theoretical analysis—we do not impose any such distributional assumption during training or evaluation. The strong empirical performance of our method across diverse datasets further validates its practical effectiveness.\n\n[1] *Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies*, JMLR 2008\n\n> **Q5:**  \"The paper only evaluates the L2 loss of the modeled field. Is this a reasonable metric? other quantities from the field (e.g., gradients, Hessians)\"\n\nThank you for the thoughtful question. We agree that evaluation metrics should reflect meaningful physical fidelity. In our experiments, we primarily report **relative L2 loss**, which is widely used and provides an **intuitive and scale-invariant measure of reconstruction accuracy**. Compared to absolute L2 loss, the relative version avoids being dominated by the scale of the underlying physical field, which can vary significantly across data.\n\nFurther, as per your request, we additionally evaluate the **relative L2 error of the gradient fields** on sea temperature. As shown in the table below, our method achieves significantly lower gradient error compared to the second-best model, Senseiver, demonstrating its superior physical fidelity.\n\n| Sea Temperature (relative L2 of gradient field) | 100 sensors | 50 sensors | 25 sensors | 15 sensors |\n| ----------------------------------------------- | ----------- | ---------- | ---------- | ---------- |\n| Senseiver + random sensor placement             | 0.5456      | 0.5470     | 0.5501     | 0.5515     |\n| PhySense + random sensor placement              | 0.4877      | 0.4884     | 0.4890     | 0.4892     |\n| Relative promotion                              | 11.87%      | 11.98%     | 12.51%     | 12.74%     |"}, {"review_id": "roY75LOre8", "reviewer": "Reviewer_dimJ", "summary": "The paper investigates physics‐sensing problems—reconstructing dense physical fields from sparse measurements—by jointly learning (i) a flow-matching generative reconstructor conditioned on arbitrary sensor sets and (ii) a geometry-aware projected-gradient algorithm that relocates sensors to near-A-optimal positions. A key theoretical result shows that minimizing the “flow loss” used in placement optimization yields a two-sided quadratic bound on classical A-optimality  along with a theorem that establishes unbiasedness of the reconstructor. Experiments on three benchmarks—channel turbulence, global sea-surface temperature, and 3-D car aerodynamics—report up to 62 % lower reconstruction error than prior art and reveal non-obvious optimal sensor layouts.", "questions": "How critical is the Conditional-Gaussian Assumption for optimality? Perhaps take a toy problem and evaluate it?\n\nEvaluate robustness or extend analysis to sub-Gaussian / heavy-tailed priors; provide empirical test with non-Gaussian toy problem?\n\nAverage absolute gradient of the reconstruction error w.r.t. each potential location, overlaid with final sensor positions. This could reveal whether the optimiser gravitates to high-information or high-variance locations.", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Coupling flow-matching reconstruction with a placement optimizer that is differentiable and geometry-constrained is novel & elegant.\n- The polynomial control between flow loss and A-optimality provides a formal link missing in some other (but not all!) sensor-placement works\n- Evals are comprehensive: 3 diverse domains (regular grid, land–sea mask, unstructured 3-D mesh) demonstrate both generality and significant gains over five baselines, including strong generative models\n- Ablation and diagnostics are good, attention visualizations build reader confidence in the design choices\n- the optimized 15-sensor sea-temperature setup achieves accuracy comparable to 100 random sensors, hinting at real cost savings in field deployments\n\nWeaknesses\n- While the manuscript demonstrates strong empirical gains, the chain that links the algorithm’s design choices to those gains is still opaque. I am unsure why it outperforms both classical A-optimal design and recent deep surrogates. \n- Theorem 3.5 proves a quadratic relation to A-optimality in expectation, but the paper never shows that low flow-loss actually correlates with low test MSE in practice.\n- The benchmarks are legitimate, but I have seen better results on these benchmarks with other classical methods (Even with QDEIM, which is a lightning fast method with good bounds)\n- Where is the physics in the method? It is being applied to fields... The `physics' itself is not utilized. The model would behave identically if the snapshots came from video frames, as long as the tensors have the same shape.", "was_revised": true, "final_justification": "Edit: After reviewing the response, and considering what is possible between today and the revision deadline, I am changing my decision from borderline accept to accept.", "rebuttal": "We sincerely thank Reviewer dimJ for providing insightful suggestions.\n\n> \"Q1: the chain that links the algorithm’s design choices to those gains is still opaque. I am unsure why it outperforms both classical A-optimal design and recent deep surrogates\"\n\nFirst, we clarify a possible misunderstanding: we do not claim to outperform A-optimal design empirically. Rather, our theoretical analysis proves a bidirectional polynomial control relationship between the flow loss and the A-optimality objective in $\\underline{\\text{Theorem 3.5}}$. Importantly, A-optimal design focuses solely on sensor placement and does not support field reconstruction, while PhySense addresses the both. Thus they are not directly comparable.\n\nBelow, we provide our detailed justification for the strong empirical gains in three aspects.\n\n**(1) Flow reconstructor has better model capacity.**\n\nInferring the full physical field from sparse observations is a highly underdetermined problem. In such scenarios, high-capacity generative models are well-suited. Moreover, compared to those deterministic models trained with MSE, generative models are better equipped to avoid the overly smooth reconstructions often seen in ill-posed inverse problems, and thus provide superior physical fidelity. \n\n**(2) Flow process enables progressive sensor placement optimization.**\n\nAs shown in $\\underline{\\text{Eq.2}}$, our sensor placement optimization is driven by the **flow loss**, formally defined in $\\underline{\\text{Eq.1}}$. A critical feature is that the flow model operates on a **noise-data mixed input $\\mathbf{X}_t$**. In contrast, non-generative models cannot support such interpolated input. They typically start from **either pure noise or a fixed initialization** (e.g., zeros), and thus lack the ability to reflect the gradual transition from low to high fidelity representations. In our case, the flow process naturally induces **a progressive sensor placement optimization across different difficulty levels of reconstruction**, corresponding to different noise levels in the flow process, which yields a more informative and tractable feedback than non-generative models.\n\nTo verify this insight, we conduct an ablation study where we remove this flow process feedback—**using pure noise instead of the noise-data mixture in the feedback loop**—and observe noticeable degradation in performance. This demonstrates that the flow-process feedback feature provides unique benefits for sensor optimization.\n\n| Turbulent Flow | 200 sensors | 100 sensors | 50 sensors | 30 sensors |\n| - | - | - | - | - |\n| PhySense + random sensor placement| 0.1233      | 0.1527      | 0.2586     | 0.5176     |\n| PhySense + **noise-start** optimized sensor placement| 0.1118      | 0.1293      | 0.1630     | 0.2286     |\n| PhySense + **mixed-start** optimized sensor placement (**origin**) | 0.1106      | 0.1257      | 0.1558     | 0.2157     |\n\n**(3) Senseiver under optimized sensor placement is still worse.**\n\nWe further compare a stronger baseline—**Senseiver with sensor placement optimized by our project gradient descent**—on the turbulent flow. Despite optimized placement, Senseiver still underperforms PhySense, especially when the sensor number is low. This highlights that while placement is important, model capacity remains a limiting factor.\n\n| Turbulent Flow | 200 sensors | 100 sensors | 50 sensors | 30 sensors |\n| - | - | - | - | - |\n| Senseiver + random sensor placement    | 0.1842      | 0.2316      | 0.3740     | 0.5746     |\n| Senseiver + optimized sensor placement **using our PGD** | 0.1388      | 0.1750      | 0.3025     | 0.4570     |\n| PhySense + random sensor placement     | 0.1233      | 0.1527      | 0.2586     | 0.5176     |\n| PhySense + optimized sensor placement  | 0.1106      | 0.1257      | 0.1558     | 0.2157     |\n\n> \"Q2: Theorem 3.5 proves a quadratic relation to A-optimality in expectation, but the paper never shows that low flow-loss actually correlates with low test MSE in practice.\"\n\nThis is a well-acknowledged property in the field of generative models. The flow loss $\\mathcal{L}_{\\text{flow}}$ quantifies the deviation of the learned velocity field from the ideal transport direction, conditioned on sparse sensor observations. A lower flow loss implies that the model has accurately learned how to map noise to the target data manifold under the given sensor conditions. As a result, samples generated under low flow loss are more likely to align with the true conditional distribution, leading to reconstructions that are more physically plausible and have lower test MSE. \n\n>\"Q3: I have seen better results on these benchmarks with other classical methods (Even with QDEIM, which is a lightning fast method with good bounds)\"\n\nThank you for this point. To the best of our knowledge, we are not aware of classical methods demonstrating better results on these complex benchmarks. QDEIM and the SSPOR we compare against share a similar idea: they both leverage POD (Proper Orthogonal Decomposition) to reduce the problem dimensionality and select sensor locations based on the dominant subspace. However, SSPOR shows limited performance in our experiments. If you may have suggestions for additional baselines, we would be happy to include them in the discussion stage.\n\nMoreover, **deep learning methods exhibit clear advantages in highly nonlinear and high-dimensional settings**. These scenarios often involve intricate spatial correlations and nontrivial patterns that classical linear subspace methods may fail to capture. Further, deep models benefit significantly from scaling up data and model capacity—just as we have seen with large language models like ChatGPT—indicating a strong potential for performance gains. Therefore, **we believe investigating deep learning–based physics sensing remains both valuable and promising**, especially for real-world applications.\n\n>\"Q4: Where is the physics in the method? The model would behave identically if the snapshots came from video frames, as long as the tensors have the same shape\"\n\nWhile we do not impose explicit physical constraints during model training, the data used in our setting already reflects underlying physical laws and thus **implicitly lies on a low-dimensional physical manifold**. This inductive structure is what makes the reconstruction process—from sparse to dense observations—both feasible and effective in physical domains. In contrast, such low-dimensional structure is generally absent in vision datasets: it is not feasible to recover a video from a few individual pixel sequences, nor to accurately interpolate intermediate frames based solely on distant snapshots. While the tensor shape ensures compatibility with deep models, it is ultimately the data distribution that determines whether a method can work.\n\nFurther, we agree that incorporating physics explicitly can be beneficial. Since it is orthogonal to our current focus, our framework is complementary to such efforts and can potentially integrate with them in future work.\n\n> \"Q5: How critical is the Conditional-Gaussian Assumption for optimality? Evaluate robustness or extend analysis to sub-Gaussian / heavy-tailed priors\"\n\nThe **Gaussian assumption is commonly used in theoretical formulations due to its analytical tractability**, particularly the ability to derive closed-form expressions for conditional distributions. This modeling choice has been widely adopted in prior works on sensor placement and experimental design [1]. Extending the theoretical analysis to sub-Gaussian or heavy-tailed priors is certainly a valuable direction; however, removing the Gaussian assumption significantly poses substantial technical challenges. We sincerely appreciate your suggestion and will consider this as an important future work.\n\nImportantly, we emphasize that this assumption is used **solely for theoretical analysis**—our method **does not impose any distributional assumptions during training or evaluation**. In fact, the empirical distributions encountered in our experiments are often more complex than both sub-Gaussian and heavy-tailed settings. The strong and consistent empirical performance across a wide range of datasets provides further evidence of the robustness and practical effectiveness of our approach.\n\n[1] *Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies*, JMLR 2008\n\n> \"Q6: Average absolute gradient of the reconstruction error w.r.t. each potential location, overlaid with final sensor positions. This could reveal whether the optimiser gravitates to high-information or high-variance locations.\"\n\nMany thanks for the insightful analysis experiment. We observe that high-variance regions tend to form continuous, band-like structures, while high-information points—measured by the gradient of reconstruction error with respect to each location—are often more discrete and spatially separated. Interestingly, **a large part of the selected sensors fall within the high-variance regions**. And **every selected sensor is found near a region of high information points, and their spatial distributions have similar patterns**. This observation suggests that the optimizer is not merely gravitating toward regions of maximal variance but is instead strategically identifying spatially separated, high-sensitivity locations, which better capture the global dynamics of the physical field. From a sensing perspective, this is quite reasonable: **maximizing overall information gain often requires sampling diverse and non-redundant locations, rather than densely covering localized variance clusters**. Due to the NeurIPS rebuttal policy, we are unable to include additional figures at this stage. However, we promise to include visualizations comparing high-information points, high-variance regions, and the final optimized placements in the appendix."}, {"review_id": "J2FUreUGL9", "reviewer": "Reviewer_ewdF", "summary": "The paper presents a framework for efficient reconstruction of physical fields. This is done by training a flow-based generative model which can reconstruct the physical field from sparse observations, and a method to select the optimal sensor readings to select based on a differentiable objective function that can be optimised with projected GD.", "questions": "1. How efficient to train the generative models in the first place, both in terms of the time required and the number of training samples or simulations that have to be used?\n\n2. How would the other sparse reconstruction methods work if a better sensor placement is used? This could potentially make the use of flow models more convincing.\n\n3. Related, why would a flow based model be required in this case? Other alternatives apart from those discussed in the paper would be things such as some encoder/decoder model (that can also quite directly generate the reconstructed field), traditional numerical solvers or even PINNs or PINOs which are known to work for inverse problems. I suspect one of the reason may be the uncertainty quantification although this is not clearly demonstrated in the presented results. Other reasons should also exist which would make the paper motivation stronger.\n\n4. Are the selected sensor placements interpretable or have some characteristics in any way? Furthermore, how sensitive are they to the GD process -- i.e., if I initialise my optimisation differently would the recovered sensor placements change?", "limitations": "Some limitations are discussed.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The proposed problem seems to have real use case and is of interesting scope. The proposed solution is also interesting.\n- The benchmarks used seem to be of sufficient difficulty (although it would also be interesting to see more realistic data used instead of simulated ones). \n- Theoretical results seem to be sufficient for the claims that are made by the paper. The link to OED methods is also a good point.\n\nWeaknesses:\n- The reported results have no error bars or confidence intervals (e.g., across the different reconstructed field cases), making it difficult to evaluate how statistically significant the results are.\n- Since the paper does also seem to focus on the sensor placement aspect, it would be interesting to see how other point selection method would work in this problem (e.g., grid placement or some configuration where the sensors are more spread out, or selection based on other ED optimality criteria), and how those placements may affect the other reconstruction methods proposed. This aspect is quite crucial to provide a more convincing and fair comparison with other methods.\n- The motivations could probably be discussed more explicitly, especially with why a quick reconstructor or even a flow-based model specifically needs to be used in this case (I don't doubt that there isn't one, but that the paper would be more complete if that was discussed).", "was_revised": true, "final_justification": "See response to the author. Mainly I think the work is likely valuable for the context of generative models, but I am less confident about comparing this work against those since I am not familiar with other works in the area.", "rebuttal": "We sincerely thank Reviewer ewdF for providing valuable feedback.\n\n>**Q1:** \"The motivations could probably be discussed more explicitly.\" \"why a quick reconstructor or even a flow-based model specifically needs to be used in this case\"\n\nSorry for the confusion. We would like to clarify our motivation further. In this work, our goal is to solve the **physics sensing task, which we define as a complete solution for understanding physical fields**. This task inherently involves two tightly coupled components: sensor placement and field reconstruction.\n\nIf one focuses solely on field reconstruction, it suffices to train a reconstructor on given observations from fixed placements. If one focuses solely on sensor placement, it may even be done using simple heuristics or statistical priors, without the need for an explicit reconstructor.\n\nHowever, our aim goes beyond these decoupled formulations. We aim to enable data-driven physics sensing—given historical or pre-collected data, we want to determine not only where to measure, but also how to reconstruct from those measurements. Therefore, the **core contribution of our framework lies in the joint optimization of sensor placement and reconstruction**, which is critical for enabling practical and accurate sensing in physical environments. \n\nPractically, PhySense is designed to output both the **optimized placement and a reconstruction model**. This enables a **closed-loop physical sensing system**: once the sensors are deployed and real-world measurements are collected, the reconstruction model can immediately infer the full physical field based on these observations. \n\nWe kindly remind this perspective may have been partially overlooked in your summary, which emphasized reconstruction but not the whole framework. And we will revise our introduction to better clarify this motivation.\n\n> **Q2:** \"Why would a flow based model be required in this case?\"\n\nThank you for the thoughtful question. Below, we provide our detailed justification in four aspects.\n\n**(1) The task we tackle is inherently generative.**\n\nInferring the full physical field from sparse observations is a highly underdetermined problem. In such scenarios, high-capacity generative models are well-suited. Notably, two of our strongest baselines, S3GM and DiffusionPDE, are also generative models. Moreover, compared to simple deterministic models trained with MSE, generative models are better equipped to avoid the overly smooth reconstructions often seen in ill-posed inverse problems, and thus provide superior physical fidelity. \n\n**(2) Flow models offer uncertainty quantification.**\n\nFlow-based models explicitly model the full conditional distribution, enabling uncertainty quantification. We reconstruct each input three times using different initial noises and then compute the standard deviation of the relative L2 error, as summarized in the table below. The results indicate a clear trend: as the number of sensors decreases, the conditional distribution becomes more diverse, leading to higher uncertainty (i.e., larger standard deviation). Further, we also calculate a pixel-wise variance map which reveals physically meaningful spatial patterns of uncertainty. However, due to NeurIPS’s rebuttal policy, we cannot include additional figures at this stage, but we promise to present pixel-wise uncertainty visualizations in the appendix.\n\n| Turbulent Flow                     | 200 sensors | 100 sensors | 50 sensors | 30 sensors |\n| - | - | - | - | - |\n| Standard deviation of relative L2 | 0.0043      | 0.0086      | 0.0210     | 0.0390     |\n\n**(3) Flow process enables progressive sensor placement optimization**\n\nAs shown in $\\underline{\\text{Eq.2 in the main paper}}$, our sensor placement optimization is driven by the flow loss, which is formally defined in $\\underline{\\text{Eq.1}}$. A critical feature is that the flow model operates on a noise-data mixed input $\\mathbf{X}_t$. In contrast, non-generative models cannot support such interpolated input. They typically start from either pure noise or a fixed initialization (e.g., zeros), and thus lack the ability to reflect the gradual transition from low to high fidelity representations. In our case, the flow process naturally induces **a progressive sensor placement optimization across different difficulty levels of reconstruction**, corresponding to different noise levels in the flow process, which yields a more informative and tractable feedback than non-generative models.\n\nTo verify this insight, we conduct an ablation study where we remove this flow process feedback—**using pure noise instead of the noise-data mixture in the feedback loop**—and observe noticeable degradation in performance. This demonstrates that the flow-process feedback feature provides unique benefits for sensor optimization.\n\n| Turbulent Flow                                               | 200 sensors | 100 sensors | 50 sensors | 30 sensors |\n| - | - | - | - | - |\n| PhySense + random sensor placement                           | 0.1233      | 0.1527      | 0.2586     | 0.5176     |\n| PhySense + **noise-start** optimized sensor placement        | 0.1118      | 0.1293      | 0.1630     | 0.2286     |\n| PhySense + **mixed-start** optimized sensor placement (**our original submission**) | 0.1106  | 0.1257  | 0.1558 | 0.2157 |\n\n**(4) Flow models are efficient enough.**\n\nThe number of training samples and the corresponding training time (in A100 GPU hours) for PhySense are reported in the table below. All baseline methods are trained using the **same dataset and number of epochs**, so the **overall training cost depends solely on model complexity, rather than whether the model is generative**. In this regard, PhySense exhibits a training cost comparable to other generative baselines. Moreover, PhySense is significantly more efficient at inference time, which maintains a relative L2 error below 0.2 with as few as 2 inference steps ($\\underline{\\text{Fig. 6c of main text}}$), significantly outperforming S3GM and DiffusionPDE.\n\n|                           | Turbulent Flow | Sea Temperature | Car Aerodynamics |\n| - | - | - | - |\n| Sample Number             | 10000          | 9843            | 75               |\n| Training Time (A100 hours) | 22             | 10              | 6                |\n\n> **Q3:** \"how other point selection method would work in this problem and how those placements may affect the other reconstruction methods proposed\" \n\nWe conducted comprehensive experiments using requested placement baselines across Senseiver and PhySense under different sensor numbers on Turbulent Flow dataset. Specifically, we consider the following strategies:\n\n- **Random**: A general baseline with no prior knowledge.\n- **Grid**: We first place sensors on a uniform grid using the largest square number less than the budget, and then randomly distribute the remaining sensors. While this strategy ensures spatial coverage, it does not account for underlying physical dynamics. In practice, its performance is comparable to random placement, which is consistent with classical sampling theory [1], suggesting that regular sampling may not outperform randomized designs in complex data.\n- **Min-Max**: A heuristic that promotes spread-out sensor placements by iteratively placing the next sensor at the point that maximizes the minimum distance to existing sensors. However, this approach does not yield significant improvements over random placement in our experiments.\n- **Our Projected Gradient Descent**: An upper-bound variant that uses the corresponding reconstruction model (Senseiver or PhySense) and performs our projected gradient descent to optimize sensor locations.\n- **Optimized Sensor from PhySense**: To evaluate the generality of the learned placements by PhySense, we feed the placement into the *Senseiver*. **The resulting performance closely matches that of the PGD and significantly outperforms all other baselines**. This suggests that our learned placements effectively capture physically informative structures that generalize beyond the inductive biases of specific reconstruction models.\n\n| Turbulent Flow | Random | Grid   | Min-Max | Our Projected GD on the reconstruction model | Optimized Sensor from PhySense |\n| - | - | - | - | - | - |\n| Senseiver-200| 0.1842 | 0.1615 | 0.1549  | 0.1388| 0.1533|\n| Senseiver-100| 0.2316 | 0.2273 | 0.2091  | 0.1750 | 0.1833|\n| Senseiver-50| 0.3740 | 0.3845 | 0.3795  | 0.3025 | 0.3035 |\n| Senseiver-30| 0.5746 | 0.5869 | 0.5747  | 0.4570 | 0.4601|\n| PhySense-200| 0.1233 | 0.1235 | 0.1196  | 0.1106 | 0.1106|\n| PhySense-100| 0.1527 | 0.1521 | 0.1533  | 0.1257| 0.1257|\n| PhySense-50| 0.2586 | 0.2436 | 0.3157  | 0.1558| 0.1558|\n| PhySense-30| 0.5176 | 0.4930 | 0.5022  | 0.2157| 0.2157 |\n\n[1] Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE TIT, 2006\n\n> **Q4:** \"Are the selected sensor placements interpretable or have some characteristics in any way?  how sensitive are they to the GD process -- i.e., if I initialise my optimisation dfferently would the recovered sensor placements change?\"\n\nFirst, the optimized sensor placements exhibit **a certain degree of physical interpretability**. For example, in $\\underline{\\text{Fig 5 of main text}}$ on the sea temperature dataset, the sensors tend to concentrate near coastal regions or ocean currents, where the physical dynamics are most active. This claim is further supported by all the showcases throughout the paper. \n\nSecond, while varying the initial placements naturally leads to different final placements, the **resulting reconstruction performance remains consistently high** across these variants, shown in $\\underline{\\text{Sec 4.4 and Fig 6b}}$. This robustness suggests that PhySense does not tailor to a specific initialization but identifies informative placements across the solution space. It also provides insight that multiple runs may help explore the Pareto optimality of placements."}]

Author rebuttal:

Meta-review: All reviewers acknowledge the excellent presentation and theotretical background. They state that:

LEF1:  The paper is "technically solid, clearly written, and impactful" 

dimJ: "Coupling flow-matching reconstruction with a placement optimizer that is differentiable and geometry-constrained is novel & elegant"  and "Evals are comprehensive: 3 diverse domains (regular grid, land–sea mask, unstructured 3-D mesh) demonstrate both generality and significant gains over five baselines, including strong generative models" 

X4JQ "The experimental section contains multiple settings covering a wide range of physical phenomena. The proposed method achieves non-trivial improvements over the existing methods in all these settings, demonstrating the effectiveness of their approach." 

All above reviewers provide a strong accept. The fourth reviewer provides a weak accept, but mentions they are not confident in their rating, as they are not from the research field. 

Given the above reviews AC clearly recommends to accept the paper. As it presents a technical novel and theoretically proven approach that yields a gain of 49%, the AC believes this paper is of significant importance to the community and is likely to make a broad impact.
Decision: Accept (oral)