FlowSymm: Physics–Aware, Symmetry–Preserving Graph Attention for Network Flow Completion
Authors: Ege Demirci
Francesco Bullo
Ananthram Swami
Ambuj Singh
Venue: ICLR 2026 Poster (2026)
Keywords: graphs
networks
flow graphs
graph attention networks
group action
bilevel-optimization
physics-aware graph neural networks
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: FlowSymm is an end-to-end graph neural model that completes missing flows by enforcing a divergence-free group-action prior, scoring corrections with attention, and refining with feature-conditioned Tikhonov regularization

Abstract: Recovering missing flows on the edges of a network, while exactly respecting local conservation laws, is a fundamental inverse problem that arises in many systems such as transportation, energy, and mobility. We introduce FlowSymm, a novel architecture that combines (i) a group-action on divergence-free flows, (ii) a graph-attention encoder to learn feature-conditioned weights over these symmetry-preserving actions, and (iii) a lightweight Tikhonov refinement solved via implicit bilevel optimization. The method first anchors the given observation on a minimum-norm divergence-free completion. We then compute an orthonormal basis for all admissible group actions that leave the observed flows invariant and parameterize the valid solution subspace, which shows an Abelian group structure under vector addition. A stack of GATv2 layers then encodes the graph and its edge features into per-edge embeddings, which are pooled over the missing edges and produce per-basis attention weights. This attention-guided process selects a set of physics-aware group actions that preserve the observed flows. Finally, a scalar Tikhonov penalty refines the missing entries via a convex least-squares solver, with gradients propagated implicitly through Cholesky factorization. Across three real-world flow benchmarks (traffic, power, bike), FlowSymm substantially outperforms state-of-the-art baselines in RMSE, MAE and correlation metrics.

Published as a conference paper at ICLR 2026
FLOWSYMM: PHYSICS AWARE, SYMMETRY PRESERV- ING GRAPH ATTENTION FOR NETWORK FLOW COM- PLETION
Ege Demirei
Department of Computer Science, UC Santa Barbara Santa Barbara, CA 93106, USA
Francesco Bullo
Center for Control, Dynamical Systems, and Computation, UC Santa Barbara Santa Barbara, CA 93106, USA
Ananthram Swami
U.S. Army DEVCOM Research Laboratory Adelphi, MD 20783, USA
Ambuj Singh
Department of Computer Science, UC Santa Barbara Santa Barbara, CA 93106, USA
ABSTRACT
Recovering missing flows on the edges of a network, while exactly respecting local conservation laws, fundamental inverse problem that arises in many systems such as transportation, energy, and mobility. We introduce FL.OWSYM,, a novel architecture that combines (i) group-action on divergence-fres flows, (ii) a graph-attention encoder to learn feature-conditioned weights over these symmetry- preserving actions, and (iii) a lightweight Tikhonov refinement solved via implicit minimum-
bilevel optimization. The method first anchors the given observation norm divergence- frre completion. We then compute an orthonormal basis for all admissible group actions that leave the observed flows invariant and parameterize the valid solution subspace, which shows an Abelian group structure under vector addition. A stack of GATv2 layers then encodes the graph and its edge features into per-edge embeddings, which are pooled over the missing edges and produce per-basis attention weights. This attention-guidee process selects aware group actions that preserve the observed flows. Finally, scalar Tikhonov penalty refines the missing entries via convex least-squares solver, with gradients propagated implicitly through Cholesky factorization. Across three real-world flow benchmarks (traffic, power, bike), FLOWSYMM outperforms state-of-the-art baselines in RMSE, MAF and correlation metrics.
of physics-
INTRODUCTION
Estimating missing flows a network is fundamental inverse problem with significant applications in transportation planning, grid reliability, and urban mobility. In practice, only fraction of the edges are equipped with sensors, and simply imputing the rest can result in physically implausible predictions. For example, this could lead traffic counts that violate conservation laws at inter- sections, power flows that overlook Kirchhoff's laws, or bike flows that disregard bike stations' capacities. Algebraically, these laws captured by the mode-balance equation B c, where B is the incidence matrix, ff the edge- flow vector, and unknown net injections. Real systems demonstrate approximate conservation rather than perfect balance due factors such noise, unobserved sources

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and sinks, and measurement errors. Therefore, a successful flow estimator must learn from the data while respecting the underlying physical principles.
1.1 RELATED WORK
Recent advances in physics-informed machine learning incorporate conservation laws via soft penal-
ties or bilevel-learned regularizers (Gao et al. 2022 Seccamonte et al., 2023 Sharma & Fink 2025
Usama et al. 2022 Wang et al. [022] Zhang et al. 2024). For example, Silva et al. (2021) learn
edge-specific regu saguaarittiin weights that trade-off flow conservation against data fideity, yielding
significant accuracy gains on traffic and power benchmarks. Other works, such as Mi et al. (2025).
embed conservation directly into pesssag--passing rules, splitting flux exchanges into symmetric and asymmetric components to ensure implicit continuity without over-constraiiing the model. These approaches balance realism (allowing some divergence) and physical consistency.
Parallel research thrusts have shown that encoding symmetries via group-equivariant Graph Neural Networks (GNNs) markedly improves sample efficiency and generalization in physical domains (Bronstein et al. 2021; Finzi et al. 2021 Pearce-Crump & Knottenbelt (2024). Architectures such as the E(n) Equivariant Graph Network (EGNN) guarantee that predictions commute with Euclidean transformations, honoring rotational and translational invariances inherent in road grids and power meshes (Satorras et al.] 2022). Hierarchical and gauge-equivariant extensions further capture multi- scale or manifold symmetries and demonstrate state-of-the-ard performance on molecular and fluid simulations (Han et al. 2022 Toshev et al. 2023). Beyond these, Murphh et al. (2019) introduced relational pooling, which aggregates representations over the permutation group to further boost discrimination power. In continuous domains, Tensor Field Networks (Thomas et al. 2018), SE(3)-
Transformer (Fuchs et al., 2020), and NequlP (Batzner et al. 2022) enforce E(3 =(u)vquivaeiance via spherical harmonics and tensor representations, demonstrating significant gains in molecular and physical prediction tasks. However, these methods primarily address spatial or geometric symmetries (invariance rotation//tanslation)) In contrast, flow completion requires respecting algebraic symmetries defined by the linear conservation constraints and sensor masks, rather than the geometric embedding of the graph.
third line of research integrates implicit differentiation and bilevel optimization to learn hyper- parameters, such as regularization strengths and group-action coefficients, end to end. Classical implicit-differentiati
techniques allow gradients to flow through optimization layers without un- rolling every iteration, greatly improving scalability. Gradient-based bilevel schemes have powered hyperparameter and meta-learning breakthroughs in deep networks (Domke 2012 Franceschi et al. 2017 2018). and recent work shows how to incorporate solver structures (c.g.. implicit time-stepping. volute flux updates) directly into GNN architectures for stable long-horizon prediction (Horic Mitsume 2022 2024). Recent works such as (mith al. (222)) and Gu al. (2020) embed
implicit physics-informed informed solvers or fixed- fixed-point equilibrium layers within GNN architectures, leverag- ing implicit differentiation to backpropagate through mass-conservation and equilibrium constraints end-to-end. Likewise, Golias et al. (2024) and Hao et al. (2023) integrate combinatorial (successive- shortest-path) and PDE-constrained solvers into bilevel optimization frameworks, differentiating through these solvers to jointly Iearn solver parameters and network weights with improved stability and efficiency. Finally, we note the growing interest in operator learning for continuous physics, such as Neural Operators (Lu et al.) 2022) and Galerkin Transformers (Cao 202T). While these models are powerful for solving PDES on continuous domains, adapting them to discrete, irregular graphs with partial observability remains distinct challenge and a complementary direction for future work.
1.2 OUR APPROACH
We introduce FLOWSYMM, a solver that augments graph attention backbone with symmetry-preservingg physics-aware corrections tailored to partially observed flow graphs. Our formulation is guided by four complementary design principles. (i) First, we reinterpret admissible divergence-free free adjustments as elements of an Abelian group action; by constructing orthonormal basis and retaining only the leading k basis vectors, we obtain a tractable latent space that scales linearly with the number of missing flows. Unlike typical equivariant GNNs, which embed spatial or permutation symmetries, our group-action basis directly spans the algebraic space of divergence-frn -free adjustments tailored to the missing flows. (ii) Second, we combine this basis with edge-wise GATv2
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embeddings (Brody et al. 2022): the attention mechanism scores each basis vector in context-awan manner, enabling the model to inject corrective flows where and when the local structure demands, while preserving flow balance the anchor manifold. This iiteetiionguided selection departs from prior bilevel diagonal regularizers by coupling missing edges through context-dependent weights over physically interpretable basis actions, rather than treating each edge independently. (iii) Third, we impose soft physical consistency through catuure-oonittiond Tikhonov refinement and train all weights end-to-end via reverse-mode implicit differentiation. (iv) Finally, we demonstrate through extensive experiments on three real-world benchmarks traffic, power, and bike that FLOWS consistently outperforms nine baselines. Our approach reduces RMSE by up ten percent while yielding attention maps and regularization weights that align with domain-specific intuition. Taken together, these contributions demonstrate that coupling learned graph attention with symmetry- preserving flow corrections offers a simple and interpretable route to state-of-ther of-t-e-rttt network-flow completion.
FLOW ESTIMATION PROBLEM
We are given a directed graph 9 = (V,E, X) with n n= vertices and m = [3] edges. Every edge e carries d-dimensional feature vector X, E Rd, stacked row-wise into X E Rmmx We frame sensor readings as noisy, partial snapshots of an underlying flow that should nearly, but not exactly, obey conservation. The goal is to reconstruct the unobserved edges while respecting these physical constraints ( specifically, flow conservation Bf = c. ) Let B E {{ {-1,0, O, 1 denote the oriented incidence matrix of G, and let E R" be the vector of net nodal injections (with o₁ for sources and C; 0 for sinks).
An ideal flow f E I.nh therefore satisfies the balance equation Bf c, reducing to classical conservation when = 0. Flows are observed on subset Eobs S E, leaving Emiss 3 \ Eobs. We denote Mobs |Eohs and mimss = -Emissl) We define binary selector matrices Satss (0, 1) Mabs X and Smiss E (0, 1 Mimics and write, for any f E I
pohn Sobaf, gmiss Smisa f, Bobs B S. Bmiss = B BTT
We lift gmnss min into RM via Siies 8miss A noisy observation f E Rmm satisfies fe fe Ee for e E Eabs and fe on Emiss- We make no explicit assumption on the statistical distribution of the noise; our reconstruction is agnostic to any particular noise model, and abbreviate the measured part as fahs = S.
Our goal is to reconstruct the missing values {fe e. E Emiss. So, we learn a mapping (g. f. + fo E RT that (i) fits the noisy measurements (small [|Sabafo - fohs fohs ||2). ). (ii) keeps the balance residual small (small BB - cll2), (iii) operates within (and respects) the admissible divergence- group-action subspace (i.e., the null-space of valid adjustments) that leaves observed flows unchanged, and (iv) couples missing edges through features X rather than a diagonal penalty.
Earlier approaches enforced hard conservation Bʃ JJi et al. 2019), used diagonal feature prior (Silva et al. 2021). or ignored the symmetry altogether. This matters because in flow completion, not modifying sensors hard operational constraint; our approach guarantees this while still exploring a rich (truncated) null-space of divergence-five -fee redistributions. This is different symmetry from Euclidean or permutation equivariances commonly studied in equivariant GNNs (Bronstein et al. 2021 Finzi et al., 2021] Pearce- Pearce-Crump & Knottenbeli (024); it is an algebraic symmetry determined by the incidence matrix together with the sensor mask (the partition observed edges and allows adjustments only on missing ones.
Ents U Emiss), which freezes
The initial estimate respects conservation (i.e., it satisfies Bʃ = e, reducing to divergence-free only when e 0). The final prediction is then allowed small, learnable violation of node balance it can handle sensor noise and better the measurements, incurring only a small residual relative to c. We provide list of symbols in Appendix
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3
OUR APPROACH: FLOWSYMM
Iuput ( (..
Prowledonal completion
Gropparttio basls
Attention-gudde selection
Solve f1
Compute proup-acien bacls ber B, Ve &oyy
(AATV2) + ( f Ual
Implici bilevel training End-to-end pemmeeer updato via ingless diferentation
Tikhonov refinmmennt fasauss dii 2/5 45
Output
Figure 1: Pipeline of FLOWSYMM: (1) construct provisional balanced completion, (2) build divergence-free group-action basis, (3) score and combine these actions via attention, (4) refine the missing flows with feature-conditioned Tikhonov solver, and (5) train all parameters end-to-end via implicit bilevel optimization.
3.1 INITIAL BALANCED COMPLETION
We start by anchoring the partially observed snapshot to the closest balanced flow so that all later updates stay within the physical manifold. Given the partially observed flow vector ʃ E R (zeros on all missing flows) and the injection vector с R" summarizing known sources (c, < 0) and sinks (c: 0), our first goal is to build a provisional full flow
f()) f + ST
miss 8(0),
B c Bobel Boba
jobs
jobs == Sobef.
We choose the minimum-norm solution via the Moore-Penrose pseudoinverse,
(0))
B misss (Bmisss Briss))
(c
Botss fobs)
pseudoinverse; when Buiss BT Bi is invertible the dagger reduces to the usual matrix inverse. In implementation we compute 5(0) with numerically stable least-squares routine (QR or SVD) to handle possible rank deficiency. We pick the minimum-norm solution among all å satisfying B = c - fobs to avoid introducing unnecessarily large adjustments on the missing edges; by construction this guarantees ( = e while leaving every observed entry intact. Although the choice of f(0) is not unique, it provides convenient anchor in Fea((c) = {f E R Bf = =c.
Here f-) denotes the
Prior work either fixes e 0 or enforces balance with global diagonal penalty; our anchor keeps the exact injection profile while requiring only sparse linear solve, providing principled starting point for learned corrections. In other words, because (()) already realizes the prescribed injections c, all subsequent valid adjustments of interest are taken from the subspace A u ker B u,
Ve Eobs J. that updates f(")) + u remain on the same affine manifold and continue leave observed entries unchanged. Hence the overall method preserves the conservation pattern defined by by design.
3.2 GROUP-ACTION BASIS
Intuitively, group action is any redistribution of the current flow that (i) maintains the nodal balance and (ii) leaves every instrumented edge untouched. First, observe that the of all group actions
A u E mm B O, ue Ve Eobs
an abelian group under vector addition. This group acts on any balanced flow f E Fbal(c) = {f
Bf = c):
Ia I f +Ua
a aER
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where U R is column-orthononen basis for A. The identity corresponds to 0; composition satisfies 9a 9 93 = 9x++9; and each element is invertible via 9n g-a. Because every column of U lies in ker B and vanishes Eohs+ the action preserves both the preseribed injection and the observed flows. In order to construct U, we note that
A = ker B n {u Ue - Ve E Eobs
whose dimension is Miiis (mins) Mimiss (n-1) for connected graph (with equality when Bmsss has row rank n- 1). In practice we truncate this space and keep only the first k basis vectors to control capacity and compute time; in our experiments, found that retaining = 256 basis vectors was sufficient to capture these variations (see Sec. 4.3) Formally, let
Pall Im BBB,
(orthogonal projector onto ker B),
and define the projector onto A by
PA Poal Pbal (Sbs Pbal Sobe Poal.
Intuitively, this formula first selects the space of all divergence-fres freee flows (using Phat) and then subtracts the specific components that would violate the sensor sensor-invariang invariance constraint. We then take U the top & singular vectors from the Singular Value Decomposition (SVD) of range C R which guarantees BU = 0 and Sobe
Unlike Euclidean-equivarine equivariant GNNs that encode rotations permutations, basis spans algebraic symmetries, zero sero-divergence moves specific to the observed/missing split, providing a customized latent space for flow completion. Note that our model learns weights for these specific, ordered physical modes, and is therefore not invariant to arbitrary rotations of the basis U. Any linear combination U a therefore belongs to A; adding a to the anchor f(0) moves strictly within the divergence-fiee free affine subspace while leaving all observed edges untouched.
3.3 AYTENTION-GUIDER GROUP-ACTION SELECTION
Having enumerated physics-valid basis, learn which directions matter for the current graph snapshot. Let E be the number stacked GATv2 layers. We apply L = 2 stacked GATv2 layers the edge features X, producing
H
GAI) E pmxd
H
Here He Rd denotes the embedding of edge and d' is the hidden dimension.
For every missing flow e E Emiss and basis index i E {l...., k} we compute a compatibility score (effectively local 'vote"); the factor |u!") modulates the action's strength edge and weights bases that do not touch e:
(.. He) 100 (1)
uy Rd trainable.
ge,t
In matrix form Q = [ge,s] E IR Menire Aggregating over missing edges and applying a softmax yields
E R
8; Memis
Lesis
EEEmlee
k
ag = softmax(s). (s):
00,5 = 1, ag,i > 0. Σ
The resulting action is A Uag = Dh= 00 (0 and the candidate balanced flow becomes
reand
(0) A. We write f geand,ohs Soha reand Since each v(() ker B and vanishes
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Eobs, retain foandd and SobaSo frand fobs, Finally, we define the candidate missing vector 5pthm = Smiss foand PM
feand
The term fe, measures how much the local context of edge e votes for the utilization of basis vector i. Each basis vector represents a global adjustment that spans multiple edges. Intuitively, the weight on this basis vector is the sum of the attention scores on its constituent edges. Summing these scores across all missing flows yields a global vector R ensuring that the final global correction A is determined by the aggregate demands local features. Applying a softmax turns s into a probability distribution 0g over the basis vectors. Figure 2 shows the histogram of these learned attention weights {<,,x) ) for the Traffic, Power, and Bike datasets (see Sec. 4.. Although every hasis vector receives some weight, most of the mass concentrates in few of them, indicating that the model focuses on small subset of group actions.
Tatfic Datano. Adantoon Wegglt
Towe Datane Waggra
38 Datmmtt We
E3000
Eswno
B00009
Figure 2: Distribution of learned attention weights og,; over k=256 basis actions on Traffic, Power, and Bike.
3.4 FEATURE-CONDITIONNE TIKIIONOV REFINEMENT
The first three steps produce a balanced candidate peand This step allows small, controlled deviation on the missing edges to absorb noise in the observations. The attention step outputs found f A
(with B f feand c and Sobs reand fobs). We employ Tikhonov regularized least-squares to cope with measurement noise and encoder uncertainty.
Even though the observations are noisy, they imply empirical imbalance è == Bobs fobe E R" which may differ slightly from the prescribed We use only to measure the empirical residual implied by noisy sensors; remains the physics target. The refinement moves toward reduced residual relative to while staying close to the candidate on the missing edges, 88
gattn = S Smisa feand
reand
E IR Mmites
gtik arg
min
| + Bobe rcand
a13
gaten
(1)
dE Riiins
divergence w.r.. empirical imhalance
Tikhonow pull toward the candidate
The strictly convex objective yields the normal equations
[ Tisss Buiss
AI mmmia Stik
X gattn
B (Bobs
(Bobe
poa c):
and the closed form
stik [iiss Bmiss + AI Xmmmiu
)-1 (2 53tta
B
Billo (Bors pra E)).
This provides inexpensive shift towards lower residual divergence. Updating only the missing components, we get fe f Ieaud + S ( (tiik - gatta), Because B Bmiss + Al symmetric positive definite (SPD), we use Cholesky (or CG) and back-propagate exact gradients via implicit differentiation. short, the refinement a single SPD solve that trades reduced empirical divergence for staying close to the attention attention-guided guided candidate on the missing edges.
3.5 IMPLICIT BILEVEL TRAINING
We tume all parameters by minimizing a held-out edge error while soock-rrpaggting exactly through the Tikhonov solver, rather than unrolling its iterations, which produces exact dyyer-graddend at minimal memory cost. All learnable parameters 0 = (GATv2 weights, A) are trained end-to-end by minimizing held-out loss while implicitly differentiating through the inner solver.

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Let {(e f(k) g(u be K-fold split: fold & provides observed edges ((k) obs a partially
masked input f(k) E RRT (zeros on emi and ground-truth flows glm RR validation
set e(k) S c(()) Let ((h) val E 10,1) x extract those coordinates.
(k)
Running the first three steps on fold yields fa (eand.(k) and 0g (kttn.(k)
s(k) (eand.(k)
Jo
Inner solver: On each fold, we solve the Tikhonov problem in equation I with fold-indexed quantities, producing 0g (tilk,(k) The associated normal equations mirror $3.4 and are omitted for brevity.
Outer loss. The validation loss averages squared error on held-out edges:
(0)
1
Σ val (6g (6)ik.(())
gutue,(k)) 1|
K
Implicit gradients. To compute VoL.al without unrolling, we use reverse-mode implicit differentiation. For each fold k, solve the adjoint system
(B (KTT (a)) miss miss
(k) = S (s 0g (k)e,()))
stilk,(k)
+ XI miiie
and then back- proppaatee through the full right-hand side
(Ba) = A oo
(kttm.(k) - (B ( I
(B obs B peand, 8
ie., account for the dependence of x, 0g (attn.(k) and f& (rand.oba,())) on 0. Accumulating over folds yields Vo Each outer update thus requires exactly two SPD solves per fold for the inner layer (one forward to obtain 5tik one adjoint to obtain w) and yields exact typer-gradients. helped by the strong convexity of the inner problem and the divergence-fir constraints imposed earlier. Additional code-level details are in Appendix
EXPERIMENTS
We evaluate FLOWSYMM on three real-world flow graphs that span transportation, energy and micromobility domains. For every dataset we (i) keep the original directed/undirected topology, (ii) normalize flow magnitudes to [0, 1]. (iii) encode the features in one-hot manner, and (iv) follow 10-fold edge hold-out protocol: in each fold disjoint subset Eobs is revealed for training, while the complementary Eval is used for validation/testing, For more detailed description of the datasets and results, please refer to the Appendix( C
Real transportation and power networks much larger, but reliable, edge-level ground truth flows typically available only on sensor-instrunee -nntrrmmented, clustered subgraphs comprising a few thousand links. Acquiring dense ground truth at city, or country-scale remains prohibitively expensive or impossible. These clustered subsets reflect the partial-observabiliyy scenario that our method addresses. To enable toppless comparison, we reuse the Traffic and Power configurations from (Silaa et al. (202T) (same sources and preparation protocol). In addition, we introduce a new Bike micromobiliry dataser built from Citi Bike trip logs (Mar 2025), which broadens the evaluation an undirected domain. Although public datasets are sparse, we provide complexity analysis in Appendix D
Traffic:
LA County road network as directed graph (junctions as nodes, road segments as edges) built from OpenStreetMap and compressed by merging non- bran-hinnhing chains; n = 5,749, m = 7,498 StrenStreetMap contributors) 2017, Silva et al.) 2021). Daily average counts from Caltrans PeMS (2018) are matched to edges, yielding measurements on 2,879 edges (38%); each edge has 18 features California Department of Transportation, 2018). 8). Measurements are moderately noisy due to sensor
issues and matching errors.
Power: Continental Europe transmission grid as an undirected graph from PyPSA-Eurt n = 2,048
buses, 2,729 lines/connectors with active power flows (MW) from a single snapshot; 14 edge features (Brown et al. 2018 Hörsch et al. 2018).
Micromobility graph from Citi Bike (Mar 2025): 1,966 active stations; undirected edges for the 2,647 most-frequent station pairs using bidirectional trip totals; flows are average daily transfers; 9 edge features (Citi Bike NYC 2025).
Bike:
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4.1 EVALUATION PROTOCOL AND BASELINES
Performance is quantified with two complementary statistics computed on the held-out edges: the moot-meansqqared error (RMSE), mean-absolus error (MAE), and the Pearson correlation coeffi- cient (CORR). Formal definitions of these metrics are in Appendix B All results in Table I are the
mean across the ten test splits.
We compare FLOWS against nine baselines spanning purely physics-based, purely data-driven, hybrid and bilevel formulations. Div enforces conservation by minimizing the mode-wise divergence with single regularization parameter A, which we select via line search on held-out validation set (Jia et al.) 2019). MLP is two-layer feedforward network with ReL.U activations in both layers, trained to predict edge flows solely from their 18/14/9-die 4/9m/mieniiio feature vectors. GCN is two- layer graph convolutional model using Chebyshev polynomials of degree 2 (Defferrard et al., 2017] Kipt & Welling 2017) and ReLU non-linearities, which incorporates both edge attributes and graph topology but does not impose any conservation constraint. To strengthen the data fataedriven baselines with recent expressive architectures, we additionally include: (i) EGNN (Satorras et al.) 2022) (two
EGNNConV layers), which is E(n)-equivariant Euclidean transformations; we apply it at the edge level via the line-graph construction with per-edge features; and (ii) GIN (Xu et al. 2018) (IWo
GINConv layers with sum aggregator), which is maximally powerful under the Weisfeiler-Lehmaa test. Both use the same tuning protocol as GCN/MLP (learning rates {10- 100 2.002.1003 10- widths
{8, 16, 32). early stopping on validation). We also consider two hybrid variants: MLP-Div applies the MLP's predictions as a prior to Div, and GCN-Div does the same with the GCN outputs; in both hybrids the regularization weight is again tuned on the validation split. We evaluated the bilevel diagonal regularizer framework of [Silaa et al. (2021), the previous state-of-the-ar instantiated with an MLP encoder (Bil-MLP) and GCN encoder (Bil-GCN).
4.2 RESULTS
Table summarizes the performance of FLOWSYMI against nine baselines, emphasizing the com- parison to Bil-GCN, the prior ofate-of-the-art Across all evaluated metrics, FLOWSYMM consistently achieves the best results, demonstrating the effectiveness of integrating physics-aware, symmetry- preserving group actions with attention-based encodings and feature-conditioned refinements.
Specifically, FLOWSSYMM shows substantial improvements over the previous state-of-the-an baseline, Bil-GCN, reducing RMSE by approximately 8% (Traffic), 10% (Power), and 9% (Bike). Similarly, MAE improvements over Bil-GCN are notable, with reductions of approximately 16% (Traffic), 9% (Power). and 12% (Bike). These results clearly demonstrate the additional predictive power provided by explicitly incorporating group actions rather than relying solely on diagonal regularizers learned via bilevel optimization. Additionally, FLOWSYMM achieves higher Pearson correlation (CORR) values compared to all baselines, indicating that predictions better capture the true underlying flow patterns. This advantage becomes particularly meaningful in real-world scenarios where understanding flow trends is essential for reliable operational decisions.
Table I: Comparison of prediction accuracy across baselines on the Traffic, Power, and Bike datasets. FLOWSYMM achieves the lowest RMSE, MAE, and the highest Pearson correlation (CORR) in all three domains. Relative to the previous state-of-the-an Bil-GCN, FlowSymm reduces RMSE by approximately 8% (Traffic), 10% (Power), and 9% (Bike), while also lowering MAE by up to 16% and improving correlation by up to 0.04.
Traffic
Power
Bike
Method
RMSE
MAE
RMSE
MAE
CORR
RMSE
CORR
Div
0.071 = 0.007
0.041 1: 0.009
0.76
0.036 0.006
0.018 :: 0.003
0.92
0.039
0.007
0.018
0.003
0.91
MI.P
0.083 0.005
0.055
+ -0.006
0.069
0.008
0.042
0.007
0.56
0.073
0.008
0.050
0.007
0.69
GCN
0.066 0.005
0.040
+ 0.005
0.067
0.008
0.045
0.008
0.64
0.057
0.007
0.039
0.007
0.72
GIN
0.065 0.006
0.038
1: 0.004
0.070
0.007
0.048
= 0.008
0.53
0.059 1 0.007
0.040
0.007
0.71
EGNN
0.065
0.036
0.007
0.009
0.65
0.038
0.006
0.75
MLP-Div
0.066 0.005
0.041 1: 0.005
0.81
0.034
0.006
0.016 1 0.005
0.93
0.037 0.005
0.019
0.004
0.90
GCN-Div
0.071 0.006
0.048 + 0.0005
0.81
0.034
0.006
0,017 + 0.005
0.92
0.037
0.007
0.019
0.004
0.90
0.038
0.006
0,011 0.002
0.033
0.016
0.93
RiLGGNN
0.062 0.005
0.034-
+ 0.003
0.82
0.029
0.006
0 0.002
0.94
0.032
0.006
0.016
0.003
0.93
SyomSymm
0.057 0.004
0.028:: 0.002
0.85
0.005 0.026
0,010
0.001
0.96
0,029
0.005
0,014
0.002
0,95
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4.3 ABLATION STUDY: BASIS SIZE& & ROLE OF ATTENTION
We conducted four ablation experiments: (i) the number of basis vectors, (ii) the effect of group- action setup, and (iii) the learning of weights on basis vectors through attention (iv) the effect of bilevel optimization. We report the RMSE on the held-out edges of each dataset (Table for each study. For (i), we sweep & E {64, 128, 256, 512] while keeping all other hyper-parameters fixed. Performance improves steadily up to k = 256 and then saturates, indicating that 256 basis vectors already capture the relevant divergence- -reegenne--fre variability. Larger bases offer additional benefit but incur higher memory cost, hence we adopt k = 256 as the default. (ii) Comparison of FLOWSYMM with Bil-GAT, the ablated variant without group-action modeling, reveals clear and consistent performance gap. These consistent improvements show that group-action basis provides a meaningful accuracy boost over a standard attention-only model, confirming that physics-informed symmetry constraints help guide more precise flow reconstructions. (iii) Lastly, we keep the basis size fixed at k 256 but remove the attention mechanism: 256 group actions are sampled uniformly at random and combined with single global weight. Compared with the full FLOWSYME YMM (k 256), this variant increases RMSE by +5% on Traffic. +11.5% on Power, and +6.8% on Bike). The gap shows the performance gains that arise from the edge-wise atlention weights. (iv) Finally, removing the bilevel optimization framework (GAT) results in a substantial degradation in performance.
Table 2: RMSE results across ablation variants. Increasing the basis size from 64 to 256 reduces error, after which it saturates. Removing the group-action module (Bil-GAT) or using uniform (non-attention) weights both increase RMSE
Variant
Traffic
Power
Bike
FlowSymm (k = 64)
0,060
0.029
0.031
FlowSymm (k 128)
0.058
0.028
0.030
FlowSymm 226)
0.057
0.026
0.029
FlowSymm (k = 5112)
0.057
0.026
0.029
Bil-GAT (no group-action)
0.062
0.029
0.032
GAT (No bilevel / /impliiit)
0.067
0,065
0.062
No-attention (ks = 256)
0.060
0.029
0.031
DISCUSSION
Our experiments demonstrate that FLOWSYMM achieves the strongest performance across three qualitatively different flow domains, improving RMSE by 8-10% over the previous offte-of-the- art (Silva et al. (02T). The gains are consistent across MAE and correlation, confirming that the symmetric, physics-aware corrections learned by the model not only reduce average error but also better reproduce the true spatial pattern of flows. The ablation study in Sed4.3 shows two key effects. First, enlarging the basis from k = 64 to k 256 steadily reduces error, then saturates, indicating that a few hundred group actions already capture the most important degrees of freedom. Second, replacing learned attention with random weights removing the group module altogether (Bil-GAT) incurs clear penalty. These trends support our design choice: attention lets the solver place corrective mass where local topology and features call for it, while the group basis guarantees that such corrections preserve the global conservation pattern. Although a systematic qualitative study beyond the scope of this work, the sparsity implied by that makes feasible to inspect individual high-scoring group actions and trace how they redistribute flow while remaining divergence- -ree. Because every action lies in ker B, these adjustments are directly interpretable as redistributions that respect physical laws. Viewed another way, FL.WWSYMM performs a meta-search over the divergence-frre free flow manifold: it first enumerates a truncated basis of admissible group actions and then uses attention to navigate and select optimal corrections. Intuitively. FLOWSYMM first enumerates a basis of admissible adjustments and then uses a GNN to learn a feature- based attention mechanism that selects optimal combination of these adjustments. This meta-search framing highlights how our model adaptively explores pwysics-preservinge preserving adjustments, rather than relying on fixed regularizer, find the best flow completion.

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5.1 LIMITATIONS
Despite the discussed benefits, there are some limitations. We train on single-shot graphs and do model temporal dynamics. Extending the approach with recurrent encoders or time-coupled group actions is promising direction. While k = 256 sufficed in our graphs, very large or highly cyclie networks may require more basis vectors, increasing memory; adaptive selection strategies could mitigate this. Additionally, we follow a randomized edge hold-out protocol, and if real sensor placements exhibit systematic biases (e.g.. clustered in urban centers), performance could degrade under different masking patterns. Furthermore, in cases where unobserved cycles are fully determined by known injections, our basis construction may overstate the degrees of freedom; however, the attention mechanism mitigates this by learning to suppress redundant basis vectors. Lastly, the inner solver is convex, but the overall bilevel training non-convex and may converge to suboptimal local minima, making hyperparameter initialization and tuning important.
5.2 FUTURE WORK
Beyond addressing the limitations above, we see two immediate extensions: (i) relax the known- injection assumption by jointly inferring the net nodal injections c alongside the missing flows, enabling operation in settings with uncertain or partially observed source/sink measurements; and (ii) model temporal dynamics by extending the encoder or group-action module with recurrent or graph-temporal architectures, that spatio-temporal correlations across successive snapshots inform more accurate imputations.
We also emphasize that FlowSymm is modular and complementary: any stronger encoder (including conservation-awarr message-pant (essage-asssing can replace the GATv2 block within our pipeline, while the group-action layer and implicity--diffeenntian basis-driver three-step strategy readily adapts to other missing-dar data
refinement continue provide the physics guarantees.
The same problems governed by linear constraints- for example, inferring heat or concentration fields in diffusion processes (via Laplacian group actions), estimating gaps in electrical potential fields, or imputing state variables in compartmental epidemiological models. Each application requires crafting an appropriate group action and solver but follows the same overall recipe.
In summary, our results suggest that modest, physically motivated modifications to the basis, attention weights, and the implicitly differentiated refinement suffice to push performance beyond more generic GNN bilevel formulations, while keeping the model transparent and controllable for practitioners in transportation, energy, and mobility planning.
5.3 ACKNOWLEDGEMENNT
This material is based upon work supported by the National Science Foundation under grant no. IIS- IIS-2229876 and is supported in part by funds provided by the National Science Foundation, by the Department of Homeland Security, and by IBM. Any opinions, findings, and conclusions or recommendations expressed in this material are those the author(s) and do not necessarily reflect the views of the National Science Foundation or its federal agency and industry partners.
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APPENDIX A: NOTATION LIST
Symbol
Meaning
n, m
Number of vertices / edges the graph.
= XX)
Input graph: vertex set V, edge set &, edge features X.
Eahs: Emisa
Observed missing flow subsets.
Mobs |obel mimias
Emiss]]
Number of observed missing flows.
miss n2 +
Dimension of full divergence- freeeeeeeeeeeeeeeeeeeeee group-action space.
B E 1-1, 1
Oriented incidence matrix of G.
Bobs B ST abs B Buiss = B BST miss
Incidence restricted to observed / missing flows.
Sobs E {0. 1 Moba Xm Smise E (0, 1) Muniza X
Selector matrices for observed missing flows. Projector onto ker B.
Pal m B J
miss- diag LeFmim E IR
Mask onto missing-edge coordinates. Partially observed flow vector (zeros on Emiss).
fobs S
Observed portion of f-
CE I
Net nodal injection (c, < 0: sources, G 0: sinks). Minimum- norm balanced completion (Bf c).
f(0) f ST
miss 8(0)
(00) Smiss!" f())
in f(0)
U E R.xk (f))
Truncated orthonormal basis for A (k: S r).
ith column of U, a divergence- basis vector. Number of retained basis vectors (default 256).
h
A {u: B ue =0 Ve Ears}
Admissible divergence-free free adjustments.
L
Number of stacked GATv2 layers.
X E IR
Edge-feature matrix (one row per edge).
H GATv2a (G, Roxd
Edge embeddings; He Rd for edge Input / hidden dimensions of GATv2. Trainable attention vector for basis Compatibility score between edge and basis i.
d, d'
wi E Rdd
Ge = (w!) Ho) luce
(1)
Q = [9ea] IR mimiss X
Matrix of all compatibility scores. Aggregated score for basis i.
8;
EeeEmin qe
Qg = softmax (8) E R-1
Attention weights over columns of U. Divergence-free freee group-action.
A ag
f0 = f(0) +
reand-
Candidate balanced flow.
80 gattn Smisa, reand
Candidate missing values after attention.
è = Boba fobs
Empirical imbalance from observations.
Learnable Tikhonov regularization weight. Tikhonov-refined missing values.
8 arg mins || B + Bobe geand ohs - + 218 88 ||2
fa fo reand + S (52k 5gttn)
Final model output.
K
Number of cross
cross-validenn validation folds.
(kr)e((k)
Ground-truth flows on held-out edges in fold
((A)
Selector matrix for validation edges in fold k:
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APPENDIX B: EVALUATION METRIC DEFINITIONS
Let Eval S & be the set of held-out edges of size m' |-val-- For each E Eval, let fe and Se denote the model's predicted and the ground-truth flow, respectively. We report four metries:
Root Mean Squared Error (RMSE):
RMSE
Σ
Sa)
V m eEEval
Mean Absolute Error (MAE):
MAE
m Σ fe fel.
nEEval
Pearson Correlation (Corr):
Corr cov(f, f) o(f)
where
cov(J. 1 m.
E - J), a(î)= = V Vm E 7/2
eEEval
and
f
1 ml
Ee Je mi E.fe.
APPENDIX C: EXPERIMENTS DETAILS
We use Los Angeles County's road network as directed graph in which vertices correspond to junctions and edges to individual road segments. The underlying topology is extracted from OpenStreetMap and then compressed by merging any maximal sequence of non-branching segments into a single edge, yielding 5 749 vertices and 7 498 directed links. Silva et al. (2021); OpenStreetMap contributors (2017) Traffic flows are given by daily average vehicle counts recorded in 2018 by Caltrans PeMS (Performance Measurement System) California Department of Transportation (2018) sensors placed on major highways; each sensor matched to its nearest compressed edge using geospatial coordinates and auxiliary sensor metadata. In total, 2, 879 edges (38% of the network) carry flow measurements. Every edge annotated with an 18-dimensional feature vector that includes: the latinude/longiude of its upstream and downstream endpoints; the number of lanes; the posted speed limit; a hot encoding of highway type (c.g., motorway, motorway link, trunk); and three graph-based statistics such as the in-degree and PageRank centrality of the source vertex, and the out-degree the target vertex. We note that, due to occasional sensor malfunctions and imperfect edge- sensor matchings, this dataset exhibits noisier measurements than Power and Bike benchmark, making flow estimation particularly challenging. OpenStreetMap is licensed under the Open Database License (ODbL) 1.0. Caltrans PeMS data publicly available under Caltrans' data sharing policy (CC BY 4.0).
Traffic:
undirected graph whose vertices are electrical buses and whose edges include both physical transmission lines and auxiliary bus connectors for generation and consumption. Active power flows (in MW) taken from single snapshot generated by PyPSA-Eur Brown et al. (2018) ; Hörsch et al. (2018), Python toolbox that solves optimal linear power-flow problems using historical load and generation data under default settings. The resulting network comprises 2,048 buses and 2,729 edges. Each edge is equipped with a 14-dimensional feature vector: for line edges, we include reactance x, resistance r. nominal capacity 8nom- binary flag indicating whether 8nom is extendable, the capital cost of capacity expansion, line length, number of parallel circuits, and optimized capacity sopt ; for bus edges, we encode the control strategy (PQ, PV, or slack) via one-hot encoding, A binary indicator distinguishes line versus bus edges, and all categorical attributes are represented with one-hot vectors Silva et al. (202I). PyPSA-Eur is released under multiple licenses: original source code under the MIT License, documentation under CC-BY-4.0 configuration files mostly under CCO-1.0, and data files under CC-BY-4.0.
Power: We the continental European transmission system
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Bike: We build a micromobility flow graph from the public Citi Bike system data for March 2025 Citi Bike (025). After discarding trips with missing station IDs and aggregating intra- usage, we create vertex for active locations (1 966 total). For every station pair (i, we sum the trip counts in both directions and retain the 2, 647 most-frequent pairs, resulting in an undirected edge set. Symmetrizing in this way removes spurious directionality driven by one-way streets while preserving signal. Edge flows are the average daily bike transfers over the month. Nine
the capacity-plannin edge features are retained: latitude/longitde of both locations, duration of trip. rideable type (classic / electric), fraction of trips by members, fraction of trips in peak hours (5-7PM and 7-9AM), and distance between locations. The data is made available for public use, though specific licensing terms are not explicitly stated. We have ensured compliance with all usage guidelines provided by Citi Bike.
Table 3: FlowSymm- Standard deviations (over 10 folds) for RMSE and MAE on each dataset.
Dataset
RMSE
MAE
Traffic 0.004
0.002
Power 0.005
0.001
Bike
0.005
0.002
Stability across folds (Table We report the standard deviation of both RMSE and MAE over our 10 cross-validation folds. Across all three datasets, FL.WWSYMM exhibits very low variability. fluctuates by only 0.004-0.005 and MAE by 0.001-0.002, indicating that its performance is highly consistent regardless of which edges are held out.
APPENDIX D: SCALABILITY AND IMPLEMENTATION DETAILS
We present detailed complexity analysis of the components of FlowSymm. The computational cost of our method is driven by the GNN encoder and the two algebraic stages: creating an initial balanced anchor and the final Tikhonov refinement. In our analysis, we simplify the notation by treating fixed-size hyperparameters as constants. Specifically, the hidden dimension of the encoder D and the number of basis vectors & = 256 do not grow with the size of the graph. Furthermore, the number of nonzero entries in the incidence matrix, nnz( is simply 2m for a graph with m edges, so we consider its complexity to be Om)).
For numerical stability, our current implementation utilizes dense matrices for the core solver steps, which dictates the overall complexity.
Balanced Anchor: Requires computing dense pseudoinverse the n x matrix Bmisa Tisse which costs (n²)) time.
Tikhonov Refinement: Involves forming and solving the Manis X m'iss normal equations. Building the matrix Biiss Buiss costs O(m and the Cholesky decomposition costs O(m
The total per-iteration cost is the sum of the encoder, attention, and solver costs. Since the GNN and attention steps are linear in m, the overall complexity is dominated by the dense solvers:
Omm + n2 + m
The complexity of the dense solvers shifts significantly in a scenario where the number of missing edges m'iis assumed to be a fixed constant that does not grow with the overall size of the network. Under this condition, any term dependent on mimis becomes constant. Specifically, the cost of the Cholesky decomposition, O( becomes a constant factor that can be disregarded in the asymptotic analysis. Similarly, the cost of forming the normal equations matrix, Omm simplifies to On)). The total per-iteration complexity, O(m + n23 + m therefore reduces to O(m + n²)). Consequently, the overall complexity is dominated by the balanced anchor step, and the practical bottleneck becomes the number of nodes in the graph, simplifying the complexity to (n²)). The dense implementation is not fundamental our method. By using iterative solvers, the scaling can be dramatically improved.
Both the anchor and refinement steps can be reformulated as sparse linear systems solvable with the Conjugate Gradient (CG) method. The O(n²) cost of the anchor step is result
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of direct matrix inversion. The CG method avoids this cubic bottleneck by re-framing the problem as an iterative solver. Each iteration requires only matrix-vecton products, which be computed in (m) time without using the dense matrix, thus reducing the overall complexity of this step to linear.
These solvers operate via matrix-vector products (e.g,, v + (Biise miss + Xf)o), which be computed efficiently without materializing dense matrices, costing only (()) per iteration.
Assuming the number of CG iterations TeG is effectively constant for well-conditioned graphs, the complexity of each solver step becomes linear. The total per-iteration cost for this ideal implementa- tion simplifies to:
(m).
This shows that FlowSymm's architecture compatible with methods that scale linearly with the number of edges, making it suitable for much larger graphs with appropriate engineering.
A APPENDIX E: ADDITIONAL EXPERIMENTAL ANALYSES
In this section, provide detailed results from three additional experiments conducted to verify the robustness, physical consistency, and architectural superiority of FLOWSYMM.
A.1
SENSITIVITY TO NOISE IN NODAL INJECTIONS (c)
key input to our method is the net nodal injection vector c, which used to compute the initial balanced anchor ((0), To evaluate robustness of FLOWSYMM to measurement errors or uncertainty in c, conducted sensitivity analysis. We trained the model using the ground-trus -trth c, and at inference time, we injected Gaussian noise into c (scaled by percentages of the standard deviation of c).
Table reeporss the L2-norm of the anchor (f) the T--norm of the learned group-action correction (A), and the final Test RMSE. The results show that while noise in e significantly perturbs the anchor, the model's learned correction A (which relies on edge features) remains stable. Consequently, the final prediction performance exceptionally robust, with negligible degradation in RMSE even 50% noise levels.
Table 4: Sensitivity analysis of FLOW under varying levels of noise added to the injection vector test time. Despite significant drift in the anchor norm, the final RMSE remains stable.
Dataset
Noise Level
Anchor Norm (1)))))12
Action Norm
| |||])
Test RMSE
Traffic
0%
17.45
5.33
0.057
15%
17.54
5.33
0.057
35%
18.21
5.33
0.058
50%
18.45
5.33
0.060
Power 0%
2.19
1.38
0.026
15%
2.19
1.38
0.026
35%
2.20
1.38
0.027
50%
2.21
1.38
0.028
Bike 0%%
2.27
1.05
0.029
15%
2.27
1.05
0.029
35%
2.26
1.05
0.029
50%
2.32
1.05
0.030
A.2 COMPARISON WITH "PREDICT-TIIENPPROIEC" BASELINE
To validate the benefit learning within the valid group-action subspace versus projecting onto it, we implemented "Predict-then--rret Project" (PnP) baseline. This baseline trains an identical GATv2 encoder to predict an unconstrained correction Rmm which is then projected onto the feasible subspace via A PAS.
As shown in Table 5 the PnP baseline performs significantly worse than FLOWSYMA and, notably, worse than the pgysics-agnostic ML.P. This confirms that learning unconstrained vector and
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Published as a conference paper at ICLR 2026
hoping for successful projection is difficult learning task with poor gradient signals, whereas FLOWSYMM's approach learning coefficients for a valid basis is far more effective.
Table 5: RMSE comparison between FLOWSYMM, physics-agnostic MLP, and the Predict-then-
Project (PnP) baseline. Lower is better.
Model
Traffic
Power
Bike
MLP (No Physics)
0.083
0.083
0,073
PnP Baseline (Projected)
0.201
0.127
0.145
FlowSymm (Ours)
0.057
0.026
0.029
A.3 ANALYSIS OF PHYSICAL CONSISTENCY (DIVERGENCE RESIDUALS)
Finally, we evaluate the physical adherence of the models by measuring the Ly-norm of the divergence residual, |B - c||2. Table 6 compares FL.OWSYMM against both physics-agnostic (MLP, GCN) and physics-aware (Min-Div, Bil-GCN) baselines.
The results demonstrate that ILLOSYMM achieves the lowest (or tied for lowest) divergence residual across all datasets. Unlike the MLP and GCN, which violate conservation laws significantly (high residuals), FLOW LYMMSYMM maintains strict physical consistency while simultaneously achieving state- of-the-art reconstruction accuracy.
Table 6: Final Divergence Residual (i cl(2) on the test set. Lower values indicate better adherence physical conservation laws.
Model
Traffic
Power
Bike
MLP
5.69
2.73
2.75
GCN
5.71
2.76
2.77
Min-Div
2.94
2.45
2.40
Bil-GCN
2.83
2.43
2.38
FlowSymm (Ours)
2.81
2.42
2.37
APPENDIX F: REPRODUCIBILITY STATEMENT
To ensure our results are fully reproducible. we provide a comprehensive set of resources. The complete source code for FLOWSYMM, including data preprocessing scripts, model implementations, and evaluation notebooks, is included in the supplementary material and will be made publicly available upon publication. We train all models with the Adam optimizer for at most epochs, using a base learning rate (outer_) 1r) of 10-2. We selected 10 epochs for our end-to-end bilevel training because in preliminary experiments the validation loss consistently platcaued by epoch across all folds, and extending beyond 10 yielded negligible gains at substantially higher compute cost. Early stopping is applied with patience 10 on the held-out loss. Our GATv2 encoder uses an 0014/9-dimensinn
input (depending on dataset), hidden size of h hide=16, num heads=4 attention heads. All results use 10-fold cross-validation (n_folds=10) All experiments were run on dual-socket Intel Xeon CPU Max 9470 server and single partition of NVIDIA H200 (18G). For the bilevel-MLP and bilNvel-GCN baselines, we directly adopted the hyperparameters provided in Silva al, Silva al. (2021). without additional tuning, match the original experimental setup. We tuned all other models by minimizing RMSE via grid search over learning rates {10", 10-1, 10-2, 10 10-33 and hidden-layer widths {4, 8, 16). For the Min-Div baseline we ran up to 3,000 iterations, and for MLP and GCN up to 5,000 iterations, employing early stopping if no improvement was seen for 10 consecutive steps. In order to reproduce the flow-completion experiments, one can begin by opening the notebook f flowpred_pipe weweeddppppeeineeee line ipynb. This file is streamlined version of the raw command line pipeline: stitches together every preprocessing step, indtiliiiiiiiiiztiinn block, and training loop in exactly the order described in this report. Once the required Python packages have been installed with pip install), no further configuration is necessary. Note that the code embedded in flowpred_ pi pipel ne. ipybb is configured to demonstrate vros-vvaldattion
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Published as a conference paper at ICLR 2026
fold so that a first pass finishes quickly; one switch to full 10-fold evaluation using ten fold function from the evaluation.
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Reviews:
[{"review_id": "vofqEahAJu", "reviewer": "Reviewer_wnzU", "summary": "This paper introduces FlowSymm, a solver that augments a graph attention backbone with symmetry-preserving, physics-aware corrections for partially observed flow graphs. This method for estimating missing flows in networks (e.g., traffic, power, bike-sharing) where only a subset of edges have sensors.\n\nIts main contributions are:\n1、It reinterprets admissible divergence-free adjustments as elements of an Abelian group action. \n2、It combines this basis with edge-wise GATv2 embeddings. An attention mechanism scores each basis vector in a context-aware manner, enabling the model to inject corrective flows where the local structure demands, while preserving flow balance.\n3、It imposes soft physical consistency through a feature-conditioned Tikhonov refinement and trains all weights end-to-end via reverse-mode implicit differentiation within a bilevel optimization framework.\n4、Extensive experiments on three real-world benchmarks show that FlowSymm consistently improves upon nine baselines, reducing RMSE by up to ten percent and yielding performance gains over the previous state-of-the-art.", "questions": "Questions：The entire method, starting with the \"balanced anchor\"  , assumes the net nodal injection vector c is known. In many real-world scenarios (e.g., unmonitored power grid feeders, unobserved traffic sources/sinks), c is also partially unknown or highly uncertain. How sensitive is FlowSymm's performance to errors in c?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This is a very well-written and clearly presented paper. The core idea—leveraging a group-action basis for divergence-free corrections in network flow completion. The methodology is explained with remarkable clarity, and the experimental results are thorough and convincing. The core innovation is the reformulation of the flow completion problem through the lens of group theory. While GNNs typically encode spatial symmetries or permutation symmetries, this work defines a new, problem-specific \"algebraic symmetry\" derived directly from the graph's incidence matrix and the sensor mask. \n\nIn addition, the mathematical derivation is sound, from the initial balanced anchor using the Moore-Penrose pseudoinverse to the construction of the projector P_A and the implicit differentiation for the bilevel problem. The method is built on a firm algebraic foundation.\n\nFinally, the experimental design is robust. The use of three distinct, real-world domains (Traffic, Power, Bike) demonstrates generalizability.", "weaknesses": "1、The introduction does an excellent job of covering related work in physics-informed ML and equivariant GNNs. However, the transition to the specific \"algebraic symmetry\" of flow conservation could be slightly more explicit for a reader unfamiliar with this background.\n2、The derivation of the projector P_A is technically correct but may be dense for some readers. The step of subtracting the projector onto the observed edges within the balanced space is crucial but could be briefly motivated with an intuitive phrase.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "V5kzIfEUEM", "reviewer": "Reviewer_ietj", "summary": "Given partial and noisy measurements on the edges of a graph, this paper addresses the problem of completing the missing measurements subject to conservation laws. This problem arises in applications such as transportation and power planning, where accurate measurements cannot be obtained for all edges. The authors’ approach has several steps. First, an initial solution satisfying conservation (with external sources) is obtained by solving a linear system. Then, this solution is refined through a learning procedure that incorporates measured features. Because the refinement is restricted to the null space of valid solutions, conservation is guaranteed to be preserved. The proposed approach is evaluated against competing methods on several tasks and demonstrates favorable performance.", "questions": "Please address the presentation issues and specific questions raised above.  \n\n**Additional minor comments and questions (not a comprehensive list):**  \n* **Line 106:** “while respecting these physical constraints” — does this refer to flow conservation constraints?  \n* **Section 3.3:** Could you please provide an explanation or intuition for how GATv2 couples the recovery of missing edges with the edge features $X$?  \n* **Line 284:** “SPD solver” — should this read “SPD solve”?  \n* **Section 4:** What are the features associated with each edge for the different problems?  \n* **Sections 4.1, 4.2:** Is it “ten baselines” or “eight baselines”?  \n* **Line 412:** “imptovements” — fix typo.  \n* **Line 449:** “meta-search over the divergence-free flow manifold” — overstated or overcomplicated; please revise.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 1, "presentation": 1, "contribution": 3, "strengths": "The problem addressed in the paper is important and grounded in real-world applications where measurements are an expensive resource. The use of neural attention for feature-driven completion is compelling. State-of-the-art results presented on three datasets demonstrate strong potential.", "weaknesses": "The biggest issue I have with this work is its **presentation**. While the exposition is mostly clear in terms of English, it feels unnecessarily complicated and often hinders understanding. As a non-expert, I found myself sifting through dense sentences overloaded with terminology, trying to distill the essence of each component of the work, including the problem itself, but especially the proposed approach. This issue is apparent as early as the abstract, which I understood better after reading the paper, yet still find quite convoluted. Ultimately, I must admit that I am not confident I fully understand the entirety of the proposed approach. I believe I have a reasonable high-level understanding of the procedure, but the presentation seems unnecessarily complex. Consequently, I believe the paper requires significant work, and I cannot recommend accepting it in its current form. \n\nLine 377 reads: *“physics-aware, symmetry-preserving group actions with attention-based encodings and feature-conditioned refinements.”* I find this statement overstated and misaligned with what I gathered from the manuscript.  \n\nTo illustrate the concerns above, and to raise a few additional ones, I would like to break down the outline of the proposed approach as presented in Section 1.2:  \n\n* **(i)** *“First, we reinterpret admissible divergence-free adjustments as elements of an Abelian group action; by constructing an orthonormal basis and retaining only the leading k basis vectors, we obtain a tractable latent space that scales linearly with the number of missing flows. Unlike typical equivariant GNNs, which embed spatial or permutation symmetries, our group-action basis directly spans the algebraic space of divergence-free adjustments tailored to the missing flows.”*  \n  To my understanding, after reading Sections 3.1 and 3.2, this step characterizes the subspace of solutions to the linear system over the missing edges, subject to the available measurements, in a form similar to $x = x_0 + Bz,$ where $x_0$ is a particular solution (e.g., minimum-norm) and the columns of $B$ form a (possibly truncated) basis for the null space of the linear system. If this interpretation is correct, I find the presentation misleadingly complicated (e.g., invoking Abelian groups and “physically informed bases”). It also raises some concerns; for instance, do exact solutions to the completion problem even exist in the presence of noisy measurements? In what sense does the null space provide a good characterization of admissible solutions in the noisy case? If, however, I have misinterpreted the proposed idea altogether, I encourage the authors to clarify and revise the text to avoid such misunderstandings.  \n\n* **(ii)** *“Second, we combine this basis with edge-wise GATv2 embeddings (Brody et al., 2022): the attention ... enabling the model to inject corrective flows ... preserving flow balance on the anchor manifold. This attention-guided selection departs from prior bilevel diagonal regularizers by coupling missing edges through context-dependent weights over physically interpretable basis actions, rather than treating each edge independently.”*  \n  My understanding is that an attention-based neural network is trained on edge features to select a solution within the subspace identified above. Again, I may be oversimplifying or misunderstanding, and I ask the authors to clarify as before.  \n\n* **(iii)** *“Third, we impose soft physical consistency through a feature-conditioned Tikhonov refinement and train all weights end-to-end via reverse-mode implicit differentiation.”*  \n  From Sections 3.4 and 3.5, my understanding is that the model is trained in an end-to-end manner on a regularized version of the formulation above, using implicit differentiation to backpropagate through SPD regularization. Please provide additional clarity on this part.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JOeRraQmQu", "reviewer": "Reviewer_MoiE", "summary": "The manuscript addresses edge-flow imputation under conservation on graphs. The proposed method comprises three key components: (1) an initial minimum-norm balanced completion that satisfies B f = c while keeping observed edges fixed; (2) a physics-aware group-action subspace for all observed edges from which an orthonormal basis U is constructed (truncated to k columns), and attention over this basis selects divergence-free corrections that cannot alter observed edges; (3) a lightweight Tikhonov refinement solved as a single SPD system with exact implicit differentiation via a Cholesky/CG solve. Experiments on Traffic, Power, and Bike benchmarks demonstrate consistent gains over a broad set of baselines, with ablations for basis size, attention, and the bilevel/implicit components.", "questions": "1.\tA single global λ seems suboptimal if the noise isn't uniform. If some parts of the network are much noisier, how does the model adapt? Dose learning a different λ for each edge provide the flexibility needed to handle different noise levels?\n2.\tMany state of the art variants of GNN are selected as baseline, however, some other physics informed machine learning model (with similar ideas) may also perform well (thus be a good baseline candidate) on such tasks, e.g. [1] (transformer with interpretable basis), [2] (neural operator with learnable basis).\n3.\tThe proposed model is claimed to have a good balance in physics and noisy data, however, the performance on data is verified on better RMSE/MAE/CORR, then what about physics? E.g. is the Kirchhoff’s law respected better in the power case？\n4.\tAre the k basis vectors sorted by singular value? Does this mean you're assuming that large-scale flow patterns are important, while small, local ones can be ignored?\n5.\tYour global attention applies one uniform fix across the entire graph. How does it handle specific local regions that have completely different or more complex physics? Does the model fail to capture these important local dynamics?\n\n[1] Cao, Shuhao. \"Choose a transformer: Fourier or galerkin.\" NeurIPS (2021): 24924-24940.\n[2] Lu, Lu, et al. \"A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data.\" Computer Methods in Applied Mechanics and Engineering 393 (2022): 114778.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. Clear physics prior with strict guarantees: the anchor solution and group-action construction ensure corrections remain on the B f = c manifold and do not alter sensor edges.\n2. Interpretability: attention over ker(B) basis elements (zero on observed edges) admits a physical reading as redistributions consistent with conservation; reported sparsity aids inspection.\n3. Efficient inner solver with exact hypergradients: the Tikhonov layer yields an SPD system amenable to stable Cholesky/CG solves; implicit differentiation avoids unrolling.\n4. Thorough baselines and ablations across physics-free, physics-soft, and bilevel methods; ablations isolate the effect of k, group actions, attention, and bilevel training. Reported gains are consistent across RMSE/MAE/CORR and across domains (traffic, power, bikes).", "weaknesses": "1. The core components of the proposed model, such as the single regularization weight and the global attention, assume uniform noise and physics, limiting its flexibility in handling more complex, heterogeneous networks.\n2. While the paper claims physics-awareness, it only evaluates data-fit metrics (like RMSE) and fails to directly measure how well the final predictions actually adhere to the physical conservation laws.\n3. The comparison could be improved by including other relevant physics-informed baselines in addition to GNNs; meanwhile, key details about the basis vector sorting method are unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JdNc7yt28y", "reviewer": "Reviewer_yn4E", "summary": "FlowSymm first anchors a balanced solution, then builds the feasible subspace defined by conservation and frozen observations, and updates only within that subspace via a group-action using learned weights over basis directions. A small Tikhonov-style refinement with implicit differentiation absorbs noise. Experiments on traffic, power, and bike datasets show improvements over baselines.", "questions": "1. Are results invariant to basis rotations within the same subspace A? Does the claimed “group action” imply basis-choice invariance?\n2.  Add a predict-then-project baseline, train a GNN to output unconstrained \\(z\\) and project \\(\\Delta=P_A z\\), to verify that gains come from the basis parameterization rather than projection alone. \n3. How do measurement noise/bias in \\(c\\) affect (i) the anchor \\(f^{(0)}\\), (ii) the subspace/basis \\(A,U\\), and (iii) final predictions? \n4. What is the learned \\(\\lambda\\) range, and how does it control the constraint residual \\(\\|Bf - c\\|\\) vs. reconstruction error?\n5. The “degrees of freedom” in §3.2 are only defined with respect to the “divergence-free + fixed observations.” But if an unknown cycle has a known injection, then the values on that cycle are actually determined. Does this overstate the DOF? Is that good or bad in practice?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Cleanly hard-enforces conservation and frozen observations via a feasible subspace; updates are interpretable as combinations of physical modes; solid gains on traffic/power/bike.", "weaknesses": "No test of basis-choice invariance; missing a predict-then-project baseline to isolate the benefit of the basis parameterization; limited analysis of noise/bias and λ trade-offs, with potential DOF overestimation when extra linear constraints exist.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)