Physics-Informed Implicit Representations of Equilibrium Network Flows

Flow networks are ubiquitous in natural and engineered systems, and in order to understand and manage these networks, one must quantify the flow of commodities across their edges. This paper considers the estimation problem of predicting unlabeled edge flows from nodal supply and demand. We propose an implicit neural network layer that incorporates two fundamental physical laws: conservation of mass, and the existence of a constitutive relationship between edge flows and nodal states (e.g., Ohm’s law). Computing the edge flows from these two laws is a nonlinear inverse problem, which our layer solves efficiently with a specialized contraction mapping. Using implicit differentiation to compute the solution’s gradients, our model is able to learn the constitutive relationship within a semi-supervised framework. We demonstrate that our approach can accurately predict edge flows in AC power networks and water distribution systems.

[1]  Martin Grohe,et al.  Solving AC Power Flow with Graph Neural Networks under Realistic Constraints , 2022, ArXiv.

[2]  M. Schaub,et al.  Simplicial Convolutional Filters , 2022, IEEE Transactions on Signal Processing.

[3]  Jinheon Baek,et al.  Edge Representation Learning with Hypergraphs , 2021, NeurIPS.

[4]  Francesco Bullo,et al.  Robust Implicit Networks via Non-Euclidean Contractions , 2021, NeurIPS.

[5]  Samy Wu Fung,et al.  Learn to Predict Equilibria via Fixed Point Networks , 2021, ArXiv.

[6]  Joan Bruna,et al.  Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges , 2021, ArXiv.

[7]  Santiago Segarra,et al.  Principled Simplicial Neural Networks for Trajectory Prediction , 2021, ICML.

[8]  Ian R. Manchester,et al.  Lipschitz Bounded Equilibrium Networks , 2020, ArXiv.

[9]  L. Ghaoui,et al.  Implicit Graph Neural Networks , 2020, NeurIPS.

[10]  J. Z. Kolter,et al.  Monotone operator equilibrium networks , 2020, NeurIPS.

[11]  Stephen P. Boyd,et al.  Differentiable Convex Optimization Layers , 2019, NeurIPS.

[12]  J. Z. Kolter,et al.  Deep Equilibrium Models , 2019, NeurIPS.

[13]  Laurent El Ghaoui,et al.  Implicit Deep Learning , 2019, SIAM J. Math. Data Sci..

[14]  Santiago Segarra,et al.  Graph-based Semi-Supervised & Active Learning for Edge Flows , 2019, KDD.

[15]  Kevin D. Smith,et al.  Flow and Elastic Networks on the n-Torus: Geometry, Analysis, and Computation , 2019, SIAM Rev..

[16]  Murray,et al.  An Overview of the Water Network Tool for Resilience (WNTR) , 2018 .

[17]  David Duvenaud,et al.  Neural Ordinary Differential Equations , 2018, NeurIPS.

[18]  Kevin D. Smith,et al.  Limited-Knowledge Economic Dispatch Prediction using Bayesian Averaging of Single-Node Models , 2018, 2018 IEEE International Conference on Probabilistic Methods Applied to Power Systems (PMAPS).

[19]  Xianfeng Tang,et al.  Revisiting Spatial-Temporal Similarity: A Deep Learning Framework for Traffic Prediction , 2018, AAAI.

[20]  Pietro Liò,et al.  Graph Attention Networks , 2017, ICLR.

[21]  Cyrus Shahabi,et al.  Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting , 2017, ICLR.

[22]  Thomas J. Overbye,et al.  Grid Structural Characteristics as Validation Criteria for Synthetic Networks , 2017, IEEE Transactions on Power Systems.

[23]  Jure Leskovec,et al.  Inductive Representation Learning on Large Graphs , 2017, NIPS.

[24]  Samuel S. Schoenholz,et al.  Neural Message Passing for Quantum Chemistry , 2017, ICML.

[25]  Angelos K. Marnerides,et al.  Short term power load forecasting using Deep Neural Networks , 2017, 2017 International Conference on Computing, Networking and Communications (ICNC).

[26]  Giacomo Como,et al.  On resilient control of dynamical flow networks , 2017, Annu. Rev. Control..

[27]  Razvan Pascanu,et al.  Interaction Networks for Learning about Objects, Relations and Physics , 2016, NIPS.

[28]  Max Welling,et al.  Semi-Supervised Classification with Graph Convolutional Networks , 2016, ICLR.

[29]  Lindell Ormsbee,et al.  Water Distribution Database for Research Applications , 2016 .

[30]  Arunesh Kumar Singh,et al.  Load forecasting techniques and methodologies: A review , 2012, 2012 2nd International Conference on Power, Control and Embedded Systems.

[31]  R D Zimmerman,et al.  MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education , 2011, IEEE Transactions on Power Systems.

[32]  James P. Peerenboom,et al.  Identifying, understanding, and analyzing critical infrastructure interdependencies , 2001 .

[33]  John A. Jacquez,et al.  Qualitative Theory of Compartmental Systems , 1993, SIAM Rev..

[34]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[35]  Ananthram Swami,et al.  Combining Physics and Machine Learning for Network Flow Estimation , 2021, ICLR.

[36]  Samy Wu Fung,et al.  Fixed Point Networks: Implicit Depth Models with Jacobian-Free Backprop , 2021, ArXiv.

[37]  Ralf Dresner,et al.  Fluid Mechanics With Engineering Applications , 2016 .

[38]  Babu Narayanan,et al.  Power system stability and control , 2007 .

[39]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956, Canadian Journal of Mathematics.

[40]  P. Frasconi,et al.  This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1 Short-Term Traffic Flow Forecasting: An Experimental , 2022 .