Title: IPSI: Enhancing Structural Inference with Automatically Learned Structural Priors
Venue: NeurIPS 2025
Accepted: yes
Keywords: Structural Inference

AI for Science

Graph Neural Networks
Abstract: We propose IPSI, a general iterative framework for structural inference in interacting dynamical systems. It integrates a pretrained structural estimator and a joint inference module based on the Variational Autoencoder (VAE); these components
are alternately updated to progressively refine the inferred structures. Initially, the
structural estimator is trained on labels from either a meta-dataset or a baseline
model to extract features and generate structural priors, which provide multi-level
guidance for training the joint inference module. In subsequent iterations, pseudolabels from the joint module replace the initial labels. IPSI is compatible with
various VAE-based models. Experiments on synthetic datasets of physical systems
demonstrate that IPSI significantly enhances the performance of structural inference models such as Neural Relational Inference (NRI). Ablation studies reveal
that feature and structural prior inputs to the joint module offer complementary
improvements from representational and generative perspectives.

IPSI: Enhancing Structural Inference with Automatically Learned Structural Priors

Abstract

We propose IPSI, a general iterative framework for structural inference in interacting dynamical systems. It integrates a pretrained structural estimator and a joint inference module based on the Variational Autoencoder (VAE); these components are alternately updated to progressively refine the inferred structures. Initially, the structural estimator is trained on labels from either a meta-dataset or a baseline model to extract features and generate structural priors, which provide multi-level guidance for training the joint inference module. In subsequent iterations, pseudolabels from the joint module replace the initial labels. IPSI is compatible with various VAE-based models. Experiments on synthetic datasets of physical systems demonstrate that IPSI significantly enhances the performance of structural inference models such as Neural Relational Inference (NRI). Ablation studies reveal that feature and structural prior inputs to the joint module offer complementary improvements from representational and generative perspectives.

1 Introduction

In many domains, dynamical systems can be understood as collections of interacting agents, ranging from physical, biological to multi-agent systems [9, 5, 15, 13, 1, 10]. These interactions are often modeled by an interaction graph, where nodes represent agents and edges denote the existence of interactions. Understanding the structure of such a graph is essential for analyzing, controlling, and optimizing the behavior of the underlying system. However, in practice, the interaction structure is frequently unobserved or only partially known, and only the observable agent states are available. For instance, in molecular biology, understanding the interactions between drug compounds and target proteins is critical for applications such as drug discovery, side-effect prediction, and drug repurposing [6]. These interactions are typically governed by underlying molecular structures and biochemical affinities, which are difficult to deduce purely from theoretical analysis. Moreover, experimentally identifying all potential interactions is often costly and time-consuming. Structural inference offers a promising alternative by uncovering latent interaction patterns based on molecular dynamics that can be observed more easily, thereby reducing the reliance on expensive experimental procedures and providing mechanistic insights into pharmacological activity [15].

As a milestone in the field of structure inference, the Neural Relational Inference (NRI) model leverages the latent space of a Variational Autoencoder (V AE) [7] to model underlying structures [8]. However, it may face challenges when applied to complex physical systems such as charged particle interactions. This is largely due to the limitations of unsupervised joint training and the reliance on overly simplistic priors for latent variables. While some approaches incorporate prior structural knowledge, few systematically address how to conveniently obtain reliable structure priors across diverse dynamical systems.

To address these challenges, we propose the Iterative Pretrained Structural Inference Framework (IPSI), an iterative framework for structural inference that combines a pretrained structural estimator SIprior with a VAE-based joint inference module SIjoint. Specifically, SIjoint receives embedding representations including structural information and a learnable structural prior both provided by SIprior. These components, together with state labels, provide multi-level supervision for the various modules within SIjoint during training. Meanwhile, SIjoint and SIprior are alternately updated in an iterative process, allowing the model to escape local optima and progressively refine its structure inference.

SIprior can be seen as a structural prior network, and is trained under supervision using structural labels. To this end, we propose two complementary strategies to generate structural labels for the first iteration, while in subsequent iterations, pseudo-labels from SIjoint are used instead. The design of IPSI follows a general paradigm common to many structure inference models, enabling seamless integration with a variety of existing frameworks. Experimental results demonstrate that IPSI significantly improves the performance of multiple baselines and achieves state-of-the-art accuracy across several datasets.

2 Related work

Structural inference aims to uncover latent interaction structures from observable sequences of agent states. Compared to traditional statistical and information-theoretic methods, recent deep learning approaches-particularly those based on neural networks-offer enhanced capabilities in modeling high-dimensional data requiring fewer assumptions. A major milestone in this direction is NRI, which pioneered the use of VAEs for inferring latent structures [8]. NRI employs a fully connected Graph Neural Network (GNN) encoder to propagate information across nodes, while modeling latent variables as probabilistic adjacency matrices. These latent structures are then used by the decoder to predict future states. However, NRI relies on oversimplified priors over the latent space and faces challenges in jointly training the encoder and decoder without external supervision-particularly in complex systems.

Building upon this foundation, some subsequent works have explored incorporating prior knowledge of real-world interaction structures into the NRI framework. Li et al. [11] and Chen et al. [2] introduced structural priors derived from real-world networks-such as degree distributions, sparsity, and connectivity-as regularization terms to guide latent structure learning. Along with these structural priors, Wang et al. [16] proposed an iterative training scheme in which edge weights are refined during training to emphasize likely interactions and suppress noisy connections. While these methods demonstrate improved structure inference, they often depend on manually crafted priors that are difficult to generalize across domains.

In addition to incorporating structural priors, several variants and extensions of NRI have been proposed to enhance performance and flexibility, including alternative decoding mechanisms [21], multi-interaction systems [22], and improved optimization strategies [4], among others [17, 23, 24, 18, 19]. Despite these advances, limited attention has been given to the question of how to obtain informative structural priors efficiently. This gap becomes critical in unsupervised settings, where the joint learning of dynamics and structure can suffer without effective prior guidance.

3 Preliminaries

3.1 Notations and Problem Formulation

In this paper, we denote the number of agents in the interacting system as N . The state of agent i at time t is represented by a vector x t i ∈ R d , where d is the dimensionality of the state space and the number of time steps is denoted as T . For simplicity, the collection of all agent states at time t is denoted as x t = { x t i } N i =1 ∈ R N × d , the time series data of agent i is denoted as x i = { x t i } T t =1 ∈ R T × d , and the collection of all time series data of all agents is denoted as x = { x i } N i =1 ∈ R N × T × d . The underlying interaction structure among the agents is modeled as a graph G = ( V , E ) , where V = { v 1 , v 2 , . . . , v N } represents the set of vertices (agents), and E ⊆ V × V represents the set of edges encoding the interactions between agents. The interaction graph is characterized by its adjacency tensor z ∈ R N × N × K to consider multiple types of interactions,

where each entry z ijk is defined as:

The problem of structural inference we consider in this paper is to recover the hidden interaction graph G or equivalently its adjacency tensor z from the observed sequences of agent states x . More formally, given the agent states over time, the objective is to learn a function f : x → z , where f ( · ) captures the underlying structure from states. This problem involves both inferring the structure of the graph G and modeling the interactions mechanisms that govern the dynamics of the system.

3.2 Neural Relational Inference

3.2.1 Model Overview

NRI is a foundational method for unsupervised structure learning in dynamical systems, which learns the latent interaction graph among agents from observed trajectories without ground-truth edge information [8].

Built on the VAE framework, NRI consists of two main components: a GNN-based encoder and a trajectory-prediction decoder. The encoder processes observable node sequences x ∈ R N × T × d through message passing to extract structural information and outputs a distribution over latent edge types, modeled as categorical variables with K classes, forming the adjacency tensor z ∈ R N × N × K :

Since directly sampling the discrete adjacency tensor z is non-differentiable, NRI employs the Gumbel-Softmax trick [12] for backpropagation. The inferred graph z is then fed into the GNN-based decoder, which predicts future dynamics via message passing on the sampled interaction graph. The decoder is optimized to minimize reconstruction loss between predicted and true future states:

The model is trained by maximizing the evidence lower bound (ELBO):

where the first term minimizes state prediction error and the second regularizes the latent space by constraining q ϕ ( z | x ) to align with the prior p ( z ) (typically uniform).

3.2.2 Limitations of NRI

Despite the elegant formulation and general applicability of NRI, it suffers from several notable limitations when applied to complex dynamical systems. First, the uniform prior imposed over the latent interaction graph often deviates significantly from the true underlying structure which leads to suboptimal inference. Second, the joint unsupervised training of encoder and decoder relies solely on trajectory reconstruction loss, which may not provide sufficiently informative gradients to uncover accurate interaction structures. Lastly, the absence of external structural supervision makes one-shot training strategies prone to premature convergence to suboptimal solutions. These limitations highlight the need for a more flexible and informed structure inference framework-one that can incorporate external structural cues to provide multi-level supervision or guidance for both the encoder and decoder, and employ iterative optimization to escape local minima.

4 Model design

4.1 Motivation and Overall Pipeline Architecture

Most VAE-based structural inference models can be summarized as follows (trajectory embedding is separated from the encoder):

trajectory data embedding -- - - - → embedding vector encoder - - - - → inferred structure decoder - - - - → predicted trajectory

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This paradigm consists of three trainable components: trajectory embedding, encoder, and decoder. However, only the ground-truth trajectories are available as supervision signals. In IPSI, we address this issue by providing pretraining inputs for the trajectory embedding and encoder modules. These two inputs are used to guide the embedding and encoder modules, and together with state labels, achieving simultaneous guidance of three trainable modules, thus effectively alleviating the difficulties caused by long chain joint training. Specifically, the pretrained structure estimator SIprior extracts features from raw trajectories. These features are concatenated with the original node embeddings and fed into the encoder of SIjoint. Since the training of SIprior is supervised by structural labels, this strategy will result in richer representations that embed structural cues.

Next, SIprior's predicted edge probabilities define an informed prior q prior ϕ ( z | x ) of structure. We replace the standard uniform prior p ( z ) in the KL divergence term with this learned prior:

While the uniform prior adopted in NRI serves as a regularization mechanism to constrain the structure of the latent space, it may significantly deviate from the true underlying distribution. In contrast, IPSI employs a learned prior that functions both as a regularizer and a soft target. This approach not only guides the early stages of training but also mitigates the adverse effect of the KL divergence loss on prediction accuracy during later training phases. Figure 1 shows a comparison between two kinds of priors and the ground truth.

A key challenge in the IPSI framework lies in how to obtain structural labels for supervising the training of SIprior, since ground-truth structures are unavailable in unsupervised structural inference. In this work, we propose an iterative training framework. In the first round of iteration, the structural labels are derived either from a synthetic meta-dataset or from pseudo-labels inferred by a baseline model, depending on whether prior knowledge of system dynamics is available (more details will be introduced in Section 4.5). In subsequent rounds of iteration, as SIjoint has been trained and achieves more accurate inference, its outputs are used as new pseudo-labels to update SIprior. Through this iterative process, both SIprior and SIjoint are progressively refined, this alternating update strategy will help the model escape from local optima and ultimately achieve performance that significantly surpasses that of the baseline models. Figure 2 shows the complete pipeline architecture of IPSI.

Given that IPSI is designed based on the general paradigm of V AE-based structural inference models, it possesses universality and can be integrated into various such models. Next, we will use the NRI model as an example to illustrate how IPSI integrates and enhances the performance of the NRI model.

4.2 Pretrained Structure Estimator

The pretrained structural estimator, denoted as SI prior , adopts the same GNN-based encoder architecture as the joint inference module SI joint , but is trained in a supervised manner to extract trajectory embedding containing more structural information and generate learnable structural priors, and then input them into the joint training model SI joint . The encoder comprises a temporal node embedding module followed by a fully connected message-passing network. To better capture the temporal dynamics of each agent's trajectory, we replace the Multi-Layer Perceptron (MLP) embedding used

Sljoint

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in NRI with a Gated Recurrent Unit (GRU) [3]:

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The final edge representation h 2 , prior ( i,j ) is then used to predict the edge type through a categorical posterior distribution:

where ϕ denotes the learnable parameters of the encoder, and x represents the full input trajectories. We use ground-truth labels a ijk corresponding to K types of edges to supervise the prediction of edge existence via the binary cross-entropy loss:

Details of the source of the structural supervision signal a ijk will be introduced in Section 4.5.

4.3 VAE-based Structural Inference Model

The VAE-based structural inference model SI joint adopts an encoder-decoder architecture similar to the NRI framework, but introduces a key enhancement by incorporating prior knowledge from SI prior . Specifically, SI joint 's encoder concatenates its own node embeddings with those provided by SI prior along the feature dimension. The GRUs in Eq. (6) and Eq. (12) and the learnable prior module in Eq. (18) are independently parameterized, as SI joint and SI prior serve distinct roles: the former includes both an encoder and a decoder for inference and reconstruction, while the latter contains only an encoder that generates structural priors to guide iterative updates. Except for concatenating trajectory embeddings from SI prior after the trajectory embedding layer, the decoder and encoder of

SI joint are identical to the baseline model (such as NRI):

It is worth noting that this modification is relatively minor but both effective and widely applicable. Since encoding raw trajectory data into latent node representations is a highly generic step, this design enables IPSI to serve as a general-purpose structural inference enhancement framework that can be flexibly combined with various existing encoder-decoder architectures.

4.4 Training with Hybrid Loss

As introduced in Section 4.1, we replace the simple uniform prior p ( z ) in NRI loss with a learnable prior q prior ϕ ( z | x ) , which serves both as a regularizer and as a soft target. This learnable prior demonstrates significant advantages over the uniform prior, and, compared to manually crafted structural prior losses proposed in various subsequent works such as sparsity and smoothness, it requires no task-specific design and avoids the difficulty of tuning numerous additional hyperparameters:

where β controls the influence of the structural prior. Similar to the modifications made in the model, this modification is lightweight yet effective, and can be readily incorporated into most V AE-based structural inference models. As it requires no modification to the base architecture and simply replaces the prior distribution.

4.5 Two Sources of Supervision for Pretraining

As introduced in Section 4.1, in order to train SI prior with structural labels without ground-truth structure of the target system, we introduce two complementary strategies that provide supervision signals under different assumptions about prior knowledge.

4.5.1 Supervision from a Synthetic Meta-Dataset with Prior Knowledge

When partial prior knowledge of the system's interaction form is available, we propose constructing a synthetic meta-dataset of labeled interactions. This dataset includes multiple simulated systems with similar dynamical properties but excludes the exact target configuration, aiming to enable SI prior to learn transferable structural motifs applicable to unseen yet related systems.

For example, in systems governed by radial forces (e.g., springs or charged particles systems), we generate synthetic trajectories from various parameterized systems with edge types such as attractive, repulsive, or null, and with different distance dependencies (e.g., constant, inverse, inverse-square). Importantly, we exclude systems matching the specific parameters of the target system, thereby simulating the following situation: "The observer is aware that interactions are radial in nature, but has no knowledge of the proportion of attractive, repulsive, or null connections, nor the precise functional relationship between force magnitude and distance."

As introduced in Section 4.1, SI prior is trained on the synthetic meta-dataset using ground-truth structural labels as supervision in the first round of iteration, and trained on the target dataset using pseudo-labels as supervision in the following rounds.

4.5.2 Supervision via Pseudo-Labels without Prior Knowledge

When prior knowledge of the system dynamics is unavailable, we first train a baseline model such as NRI and use its structure inference outputs as pseudo-labels to provide supervision. Although this approach does not incorporate the additional information from the synthetic meta-dataset, the pseudo-labels still offer a more informative prior estimate than uniform prior. Furthermore, the subsequent iterative process described in Section 4.1 is applied in the same manner, alternating the optimization of modules SI prior and SI joint , ultimately leading to improved performance.

5 Experiments

5.1 Datasets

To evaluate the effectiveness of the two supervision strategies, we focus on the three synthetic physical systems proposed in [8]-the spring, charged particle, and Kuramoto oscillator systems-as our synthetic meta-datasets are constructed based on the spring and charged particle systems. We adopt the same simulation configuration as in the original setup: each system is simulated for 5000 time steps, with 5 or 10 interacting objects, and the interaction graph is set to undirected. The data is split into training, validation, and test sets with a 5:1:1 ratio. And the details about the synthetic meta-dataset are described in supplementary materials.

5.2 Baselines

We compare our model with several recent structural inference approaches that focus on physical systems with available and complete codes:

NRI [8]: a VAE-based structural inference model that jointly learns the relations and dynamics.

SUGAR [11]: a method that introduces structural prior knowledge for structural inference.

MPM [2]: a method that combines a relation interaction mechanism and a spatio-temporal message passing mechanism.

iSIDG [16]: a method that iteratively updates the encoder structure based on the inferred structure.

Our model is evaluated in both the with prior (w/ prior) and without prior (w/o prior) situations, which correspond to different supervision methods as described in Section 4.5. And the evaluation metric is edge classification accuracy, which reflects how accurately the inferred edge types match the ground truth. Due to space limitations, the results on trajectory prediction error are provided in the supplementary materials. We apply IPSI framework on two different models, NRI and MPM, and report the improved performance to evaluate the generality and flexibility of IPSI.

5.3 Results

The experimental results are summarized in Table 1. Across three types of physical systems, the performance of models enhanced with IPSI is significantly better than the original model. This improvement is particularly pronounced on the Charged Particle dataset, where the interactions include both attraction and repulsion, leading to more complex motion patterns.

Table 1 also presents the performance of SI prior trained exclusively on the synthetic meta-dataset. While SI prior outperforms the NRI baseline on charged particle datasets, it is still surpassed by the full IPSI pipeline, indicating that SI joint actively refines the initial priors through joint training rather than merely replicating them. Moreover, on datasets with lower baseline accuracy, IPSI performs better when w/ prior, highlighting the complementary benefits of the two supervision strategies. Finally, despite the non-radial nature of interactions in the Kuramoto system, SI prior still provides a reasonable structural estimate, demonstrating its generalization capability. This suggests that constructing a larger synthetic meta-dataset to train a more powerful SI prior -inspired by the scaling principles of large language models-could be a promising direction for future work.

5.4 Additional Evaluation on the DoSI Benchmark

To further assess the generalization ability of our approach under different graph structures, we conducted additional experiments on the DoSI benchmark [20], which simulates dynamical trajectories over empirically-derived graphs from real-world domains. In particular, we evaluated three biologically inspired datasets, each containing 15 nodes: Brain Networks (BN) , Gene Regulatory Networks (GRN) , and Vascular Networks (VN) . Besides the conventional Springs simulator, we additionally used the NetSims simulator, which models brain activity by assigning nodes to brain regions and edges to their interactions [14].

For this study, we applied our method under the IPSI (w/o prior) configuration. Table 2 reports the AUROC scores on all six datasets. We observe that IPSI consistently outperforms the baseline methods (NRI [8], MPM [2]) across both simulators, achieving state-of-the-art performance among

VAE-based structural inference models. These results suggest that IPSI remains robust in a more realistic evaluation environment.

5.5 Training Dynamics

To further elucidate the training dynamics, we track three metrics during a single run of SI joint training: the state-prediction mean squared error (MSE), the KL-divergence regularization term, and the edge-prediction accuracy on the training set. As illustrated in Figure 3a, the KL term initially decreases-indicating that the model closely follows the pretrained prior-then rises as SI joint diverges to learn more precise structures. In contrast, both MSE and edge accuracy improve steadily throughout training. This behavior suggests that the model first undergoes a phase of prior imitation and subsequently refines the structures based on the initial priors, resulting in enhanced predictive performance. There three experiments are conducted on charged particle dataset with 5 objects and w/o prior situation.

5.6 Robustness

Figure 3b illustrates the impact of SI prior accuracy and iteration round on SI joint performance. The results show that even when SI prior is completely untrained (guessing randomly with two edge types will achieve about 50% accuracy), SI joint still achieves the same performance as the NRI baseline. This demonstrates the robustness of the IPSI framework: even when the prior inputs are poor, IPSI is capable of learning to shield against these erroneous priors, maintaining performance comparable to the original baseline model. Meanwhile, the inserted figure shows that the accuracy steadily increases and eventually converges, demonstrating the effectiveness and stability of the proposed iterative training strategy.

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5.7 Perturbation Analysis

To gain deeper insight into IPSI, we performed a random-direction perturbation analysis: For both trained NRI and IPSI models, all model parameters were flattened into a single vector, a random unit vector of identical dimension was sampled as a perturbation direction, and scaled perturbations were added to evaluate the change in loss. We analyzed two loss functions: (1) the training loss (ELBO objective), and (2) the target loss (1-ACC), reflecting structural inference quality. Across 100 random directions, we computed two quantitative measures: non-optimal proportion (the fraction of perturbation directions yielding lower loss than the original model, within a tolerance) and relative improvement (the mean percentage decrease of loss among those non-optimal directions).

Experiments on the GRN dataset (Springs, 15 nodes, w/o prior) yielded the results summarized in Table 3. Under the training loss, both NRI and IPSI exhibit comparable local optimality, suggesting that the Adam optimizer effectively finds stable basins for both. However, under the target loss, IPSI demonstrates more stable minima, reflected by lower non-optimal proportion and smaller relative improvement. The perturbation line plots in Figure 3c further visualize these differences, where each curve corresponds to a random perturbation direction. For target loss sensitivity of IPSI (bottom right), most curves are monotonically increasing, while target loss sensitivity of NRI (bottom left) shows irregular patterns with many curves decreasing under perturbation. These observations suggest that both models converge to local optima under the training loss, but IPSI converges to better optima under the task-relevant target loss. IPSI effectively reshapes the loss landscape through iteratively updates trajectory embeddings and structural priors, thereby aligning the training objective with the structural inference target.

5.8 Ablation Study

To analyze the contribution of each pretrained input component, we perform an ablation study on its two injection points within SI joint : (1) the embedding vector and (2) the structural prior . Three reduced variants are evaluated: one using only the embedding vector, one using only the structural prior, and one using neither-equivalent to an NRI variant with a GRU embedding scheme. Experiments are conducted under the w/ prior setting on the charged particle dataset with five objects, and results are shown in Table 4.

Both single-input variants outperform the baseline without pretrained inputs, demonstrating that each component independently improves performance. The full model, combining both the embedding vector and the structural prior, achieves the best accuracy, indicating their complementarity. These findings are consistent with our design rationale: the embedding vector provides enriched representations informed by global structural cues, while the latent prior serves as a regularization signal that steers the latent distribution toward plausible interaction patterns. Together, they enhance the structural inference at both the representation and generative levels.

6 Conclusion and Limitations

In this work, we introduced IPSI, an iterative pretrained structural inference framework that alternates between data-driven inference and structure-guided refinement. Through extensive experiments on both synthetic and empirically derived benchmarks, IPSI consistently improved structural accuracy and generalization over existing V AE-based methods. The perturbation analysis further demonstrated that IPSI enhances the alignment between training and task objectives by iteratively reshaping the loss landscape, providing an intuitive explanation for its superior performance.

A key limitation of IPSI lies in the increased computational overhead introduced by its iterative process, which makes it challenging to evaluate on large-scale systems (involving dozens or more agents) and real-world datasets. Generalizing structural inference to large graphs remains a major challenge, primarily due to the quadratic growth of computational cost with respect to the number of nodes when employing fully connected GNN encoders.

Nevertheless, IPSI offers a promising direction to address this issue. By leveraging the pretrained structural prior to guide the construction of encoder connectivity-rather than using it solely as a latent prior-the effective edge complexity can be reduced from O ( N 2 ) to O ( N ) for systems with sparse underlying structures. In future work, we plan to explore scalable variants of IPSI and apply it to large-scale, real-world dynamical systems such as neural and biological networks, as well as further investigate theoretical properties of its iterative optimization process.

Acknowledgments

Code is available at https://github.com/blackbird1729/IPSI .

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E. Webb, B. Day, H. Andres-Terre, and P. Lió. Factorised neural relational inference for multi-interaction systems. arXiv preprint arXiv:1905.08721 , 2019.

Y. Yang, B. Feng, K. Wang, N. E. Leonard, A. B. Dieng, and C. Allen-Blanchette. Behaviorinspired neural networks for relational inference. arXiv preprint arXiv:2406.14746 , 2024.

S. Zheng, Z. Li, K. Fujiwara, and G. Tanaka. Diffusion model for relational inference. arXiv preprint arXiv:2401.16755 , 2024.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe problem addressed is that of full identification of the hidden interaction graph (or the adjacency matrix) of a dynamical system, which can be theoretically intractable or experimentally impractical. The authors propose a framework for structural inference in systems of interacting agents, by providing more supervision signal to the model during training. Specifically, the authors provide inputs for the trajectory embedding and encoding modules typically found in VAE-based structural inference models. Results on physical systems, along with ablation studies, are presented and discussed.\n\nSTRENGTHS AND WEAKNESSES:\n## Quality\nThe proposed approach and the problem addressed are very interesting. I find the bootstrapping idea particularly interesting. The improved performance over the baselines is noticeably compelling, which is expected given the added prior information.\n\nThat said, the practicality of the preBoost-SI method remains unconvincing. It requires more supervision, thereby taking away one of the main advantages of conventional NRI. For instance, we require a baseline model or meta-dataset for supervision of $SI_{prior}$, which is not guaranteed to be accessible for arbitrary use cases.\n\n## Clarity\nI find that the clarity of this paper is lacking in many ways.\n\n- L116. You mention joint unsupervised training of encoder and decoder. But as I understand, only the encoder is unsupervised, while the decoder has access to ground-truth trajectories for future state prediction, which would make it self-supervised. Could you clarify this?\n- Eq 18, do the learnable prior $q^{prior}$ and $q$ share the same weights $\\phi$ ? I'm assuming that's not the case, but it should be clarified. Similarly, I assume that the GRU used in Eq 12 is different from the one used in Eq 6. That is not clear.\n- L173 conflicts with L159. In L159, it should be clearer that $SI_{joint}$ is not simply an encoder. Essentially, only its encoder part coincides with $SI_{pior}$.  \n- The notations in the supplementary material severely diverge from those in the main text, making it rather impossible to follow.\n\nMinor concerns (no effect on rating):\n- Eq (6-9). I'm assuming the superscripts 1 and 2 indicate the hidden layer?\n- Eq (10,12). \"prior\" in italic\n- No clear Conclusion in the manuscript\n\n## Significance\nThis work is significant for the community, as it shows how beneficial adding physical priors can be. It provides a good review of NRI, and researchers and practitioners stand to benefit from it.\n\n## Originality\nI am not very familiar with the \"structural inference\" approach to learning dynamical systems, but to the best of my knowledge, the work presented here is original and differs significantly from that provided in the presented literature.\n\nQUESTIONS:\nMy concerns and questions mainly concern the methodology. I am happy to increase my score if most of them are satisfactorily addressed:\n1) L151: How do you show that the model escapes from local optima?\n2) L147: Failure modes haven't been investigated. Does providing wrong structural labels (bad meta-datasets or terrible pseudo-labels) significantly hurt performance? \n3) L155: What do you mean by \"it possesses universality\"?\n4) L159: Why do SI_prior and SI_joint adopt the same GNN-based encoder architecture? Is there anything discouraging users from changing the architectures?\n5) L202: The supervision from synthetic meta-datasets appears impractical on real-world problems, but very interesting. Have steps been taken to guarantee that no simulated system is present in the test set?\n6) L220: In the bootstrapping strategy, updates of $SI_{prior}$ and $SI_{joint}$ are **alternating**. Is there any advantage the alternating minimization strategy offers over ordinary **joint** minimization, as done in [1] ?\n7) Figure 3a highlights interesting training regimes. But it should be repeated over several trials for definitive conclusions to be drawn!\n\n\n\n### References:\n- [1] Nzoyem et al., Neural Context Flows for Meta-Learning of Dynamical Systems, ICLR 2025.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper points out that fixed priors of the interaction graphs limit the accuracy of neural relational inference. It proposes Pretraining & Bootstrapped Structural Inference (PreBoot-SI), a bootstrap iteration framework for structural inference. PreBoot-SI estimates the prior by training an auxiliary network on manually constructed meta-datasets with similar dynamics and known interactions. The prior module and the inference module are jointly trained to promote the structural inference accuracy. Experiments on classical synthetic datasets validate the effectiveness of PreBoot-SI.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nS1. The motivation is clear. The writing is fluent. The formulas are well-organized.\n\nS2. Training a structural prior network on meta-datasets is novel for structural inference.\n\nS3. The experimental results on three synthetic datasets are significant. The ablation studies verify the necessity of the proposed components.\n\n**Weaknesses**\n\nW1. The comparison methods are not up to date. Please consider adding more recent methods for comparison. Here are some relevant papers with no conflicts of interest.\n\nReferences:\n\n[1] Guided Structural Inference: Leveraging Priors with Soft Gating Mechanisms. In ICML, 2025.\n\n[2] Structural Inference of Dynamical Systems with Conjoined State Space Models. In NeurIPS, 2024.\n\n\nW2. PreBoot-SI is only tested on three classical synthetic datasets. Though it is sufficient for proof of concept, testing PreNoot-SI on real-world datasets or more diverse benchmarks [3] will make the results more convincing.\n\nReferences:\n\n[3] Benchmarking Structural Inference Methods for Interacting Dynamical Systems with Synthetic Data. In NeurIPS Datasets and Benchmarks Track, 2024.\n\nQUESTIONS:\nQ1. Constructing the meta-datasets requires knowing essential knowledge about the data generation process, which is a strong prerequisite for PreBoot-SI compared with existing methods that only assume minimal prior knowledge about the edge distribution. Can you discuss the applicability of PreBoot-SI to real-world systems when the data generation process is less understood, or can you figure out some real-world scenarios that match the assumptions of PreBoot-SI?\n\nQ2. How will the configuration of the meta-datasets affect the performance of PreBoot-SI? For example, the choice of hyperparameter lists of p and alpha, the discrepancy between the meta-datasets and the true data, etc.\n\nQ3. Please add more recent methods for comparison. See W1 for details.\n\n**Other Comments.**\n\nC1. This paper is essentially learning a structural prior network to infer pseudo labels for the structure given the trajectories. I wonder if the term \"structural prior network\" is more accurate than the somewhat vague term \"structural priors\" that seems irrelevant to the observational state series.\n\nC2. The caption of Figure 3 states that there are three experiments, but only two labeled subfigures are presented. Specifically, a small figure on the SI_joint accuracy against bootstrap round is embedded in Figure 3(b). Please consider better organizing Figure 3.\n\nLIMITATIONS:\nConstructing the meta-datasets requires knowing essential knowledge about the data generation process, which may not always be available. Hence, it is unclear if such a method is always feasible for real-world systems where the data generation process is less understood.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper presents PreBoot-SI, which is a framework that designs for structural inference inspired by Neural Relational Inference. The authors propose an empirical Bayes type of approach to iteratively estimate the latent/underlying structure. In the experimental study, the authors present the performance on three synthetic physical systems: spring, charged particle, and Kuramoto oscillator. PreBoot-SI demonstrates stronger performance than baseline methods.

STRENGTHS AND WEAKNESSES:
Strengths: 
1. This topic is clearly significant for the structural inference problems. 
2. The proposed method with a learned prior (empirical Bayes style) is indeed likely to improve performances. 

Weaknesses: 
1. The experimental study is not extensive to support the claim. The reviewer would love to see the result working on a larger and more complex dataset. 
2. The novelty can be very limited.

QUESTIONS:
1. What does the term Bootstrap mean here? Bootstrap typically refers to the procedure to resample/subsample the data and repeat the inference process to get uncertainty estimates (for example). The reviewer didn't see where Bootstrap appears in this paper. This is a huge concern from the reviewer. The reviewer believes this work is closer to EM or empirical Bayes type of approach (but not Bootstrapping). 

2. May the authors shorten the paragraph to describe NRI and move the model design forward?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-09

comments: SUMMARY:
The paper proposes PreBoot-SI, a bootstrapped framework for structural inference in interacting dynamical systems. The method integrates a pretrained structural estimator and a VAE-based joint inference module, which are alternately updated to refine inferred interaction graphs. The framework improves performance over existing methods (e.g., NRI, MPM) on synthetic physical systems like charged particles and springs, with ablation studies demonstrating the complementary benefits of structural priors and learned embeddings.

STRENGTHS AND WEAKNESSES:
## Strengths
- Structural inference is a foundational challenge in domains like molecular biology and physics.
- The paper is well-structured and easy to follow.

## Weaknesses
- The novelty of this work is limited: The core idea of combining pretrained priors with iterative refinement is a methodological improvement rather than a conceptual breakthrough. Similar strategies (e.g., self-training, pseudo-labeling) are not brand-new in the literature [1]. 
- Narrow evaluation scope: Results focus on synthetic physics datasets (e.g., Kuramoto oscillators), with no validation on real-world biological or multi-agent systems. This limits claims about practical utility.
- The scalability is concerned: The framework’s computational overhead (e.g., iterative training) and quadratic complexity (due to fully connected GNNs) hinder applicability to large-scale systems, as acknowledged in Section 6. For real-world large-scale dynamical systems, the scalability of the proposed approach is of concern.

[1] Iterative Structural Inference of Directed Graphs. NeurIPS 2022.

QUESTIONS:
See the Weaknesses part above.

LIMITATIONS:
The authors discuss the limitations of this work.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-09

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims to identify the internal relational structure of dynamical systems and predict their evolution in time. The authors use a graph-based architecture to infer a variational latent space inspired by VAEs which encodes those relations from trajectory data, in a two stage procedure. First, a prior is trained based on labelled latent data or pseudo-labels. Then, a second network is trained based on the structural information learnt by the prior network. This provides a significant advantage over only training in a single step without latent labels, as previous works (NRI). The method is tested over multiple trajectories in three datasets (spring-like dynamics, charged particles and Kuramoto system) showing an improvement over the baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n* The technique is valid with both labelled and unlabelled data in the latent space.\n* The method is tested over 4 different baselines and ablation studies.\n* The code is provided (not publicly available yet).\n\nWeaknesses\n* The results are modest with respect to the base architecture (MPM).\n* Usually in real world applications the latent space labels are unknown, and in that case (supervision via pseudo-labels) the method is quite pointless even though the results are marginally better than the vanilla architecture.\n* There is no mention about the computational overhead of the iterative process and the addition of an extra network.\n* The study is restricted to small systems with constant edge interactions ($\\sigma$) and fixed graph topology.\n\nQUESTIONS:\n* The prior distributions scales with O(N^2). The paper only explores two size of systems: 5 and 10 bodies. How scalable is this approach to bigger systems and sparser interactions? For example, a long chain with many bodies in it. Do the authors have an intuition or maybe have tested this method in bigger settings?\n* Have the authors tried to use an MLP instead of a GRU for the node embeddings, as in the original NRI paper? Is this an important architectural choice? \n* What if the pseudo-labels are poorly estimated? Then the joint network would have bad labels and might result in a decrease in accuracy. How difficult is to train this pipeline?\n* Line 131: \"...from raw trajectories, These features...\" might refer to \"...from raw trajectories. These features...\".\n* Equation 5: Closed parenthesis is missing.\n\nLIMITATIONS:\nLimitations are addressed appropriately.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-11"}