design liquid networks

To address the limited expressive power and high communication overhead of discrete-time graph neural networks (GNNs) in multi-agent systems—particularly under variable communication ranges, non-instantaneous communication, and large-scale deployment—this paper proposes the Liquid Graph Neural Controller (LGTC), a continuous-time GNN-based distributed controller. LGTC integrates liquid time-constant (LTC) dynamics with graph-structured modeling and leverages contraction theory to rigorously guarantee closed-loop stability. It derives a closed-form analytical solution for state evolution, eliminating the need for numerical ODE integration and significantly reducing the dimensionality of communicated variables. To our knowledge, LGTC is the first continuous-time GNN architecture designed specifically for distributed cooperative control. We theoretically prove that its contraction rate is analytically maintainable. Experiments on formation control tasks demonstrate that LGTC substantially outperforms discrete GNNs (e.g., GGNN) in stability, control accuracy, and communication efficiency.

Existing electro-equivalent circuits (EECs), liquid time-constant networks (LTCs), and their saturated variants suffer from limited generalizability, accuracy, and biological interpretability; meanwhile, gated RNNs face inefficiency and non-differentiability issues. To address these limitations, this paper proposes liquid resistance–capacitance networks (LRCs), a novel neural differential equation model that integrates circuit-theoretic priors with liquid time-constant dynamics. LRCs introduce a first-of-its-kind *liquid capacitance* mechanism to suppress oscillations, enhance stability, and improve modeling fidelity. We further derive the lightweight LRC unit (LRCU), which achieves high-accuracy, differentiable, and interpretable temporal modeling via a single-step explicit Euler discretization. Evaluated on multiple time-series benchmarks and an autonomous driving lane-keeping task, LRCs/LRCU consistently outperform state-of-the-art neural ODEs and gated RNNs in prediction accuracy, computational efficiency, and neurodynamical interpretability.

This study addresses the challenge of achieving optimal estimation of dynamic latent variables from asynchronous, irregularly timed observations in a decentralized agent network lacking shared clocks, common models, or retrainable weights. The work establishes, for the first time, that under fixed-weight bases, optimal estimation necessitates two conditions—adaptive timescale modulation and observation-gap awareness—and that this requirement is independent of network capacity. To fulfill both criteria, the authors propose a multi-timescale modeling framework grounded in continuous-time liquid networks, integrating adaptive filtering with gap-sensitive mechanisms. Empirical results demonstrate that the proposed architecture simultaneously satisfies both theoretical conditions and attains optimal estimation performance, whereas LSTM-based approaches or conventional continuous-time filters meet only one of the two requirements.

This study addresses the limitations of traditional RNNs and LSTMs in modeling continuous-time sequences with sparse or missing data, such as clinical physiological signals. It systematically evaluates liquid neural networks (LNNs), particularly the Closed-form Continuous-time (CfC) model, on multimodal native time-series tasks by modeling hidden-state dynamics through continuous differential equations and incorporating temporal dropout for robustness stress testing. Experiments across diverse datasets—including N-MNIST, QuickDraw, IAM, and PhysioNet Sepsis-3—demonstrate that LNNs significantly outperform LSTMs in both parameter efficiency and resilience to missing data. The results highlight the superior practical utility of LNNs in real-world applications such as event-based vision, handwriting recognition, and clinical monitoring.

This study addresses the prediction of energy budgets during droplet impact and coalescence under surface-tension-dominated multiphase flow. We propose a two-stage temporal modeling framework: first, an LSTM network directly predicts the dynamic evolution of kinetic, dissipated, and surface energies using easily measurable geometric time-series data (e.g., droplet diameter); second, key dimensionless numbers (Reynolds *Re* and Weber *We*) are inversely inferred from predicted energy trajectories to bridge simulation and experiment. Our approach is the first to couple geometric observations with energy conservation principles—requiring no velocity or pressure field inputs. Validated across a broad *Re*–*We* regime, the model achieves <5% error in all three energy components. It demonstrates strong generalizability across operating conditions and enables direct transfer to experimental datasets.

Multivariate time series often exhibit heterogeneous dynamics across multiple scales and irregular patterns, which are challenging for conventional models to capture effectively. This work proposes a Multi-Rate Mixture of Experts (MR-MoE) framework that, for the first time, integrates multi-timescale expert decomposition with Liquid Neural Networks (LNNs). MR-MoE employs a gating mechanism to enable input-adaptive expert specialization and incorporates both feature-level and temporal attention modules to enhance modeling capacity and interpretability. Evaluated on multivariate time series forecasting tasks, MR-MoE significantly outperforms LSTM, monolithic LNN, and standard MoE baselines in terms of AUROC and AUPRC metrics, while maintaining favorable computational efficiency.

This study investigates the impact of intrinsic multiscale temporal dynamics on grid cell coding within path integration. To this end, leaky dynamics are introduced for the first time as a low-pass filtering mechanism into recurrent neural networks (RNNs), yielding a continuous attractor–based leaky RNN capable of adaptive timescale modeling. This approach stabilizes network dynamics and facilitates the emergence of regular hexagonal grid activity patterns alongside toroidal attractors with a central hole. Compared to conventional RNNs, the proposed model achieves substantially higher positional estimation accuracy and demonstrates enhanced dynamic robustness and more stable grid representations, even under noisy conditions.