Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling

Integrating physics models within machine learning models holds considerable promise toward learning robust models with improved interpretability and abilities to extrapolate. In this work, we focus on the integration of incomplete physics models into deep generative models. In particular, we introduce an architecture of variational autoencoders (VAEs) in which a part of the latent space is grounded by physics. A key technical challenge is to strike a balance between the incomplete physics and trainable components such as neural networks for ensuring that the physics part is used in a meaningful manner. To this end, we propose a regularized learning method that controls the effect of the trainable components and preserves the semantics of the physics-based latent variables as intended. We not only demonstrate generative performance improvements over a set of synthetic and real-world datasets, but we also show that we learn robust models that can consistently extrapolate beyond the training distribution in a meaningful manner. Moreover, we show that we can control the generative process in an interpretable manner.

[1]  Laura von Rueden,et al.  Informed Machine Learning – A Taxonomy and Survey of Integrating Prior Knowledge into Learning Systems , 2019, IEEE Transactions on Knowledge and Data Engineering.

[2]  David A. Moore,et al.  Embedded-model flows: Combining the inductive biases of model-free deep learning and explicit probabilistic modeling , 2021, ICLR.

[3]  Ufuk Topcu,et al.  Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling , 2021, ArXiv.

[4]  Sofie Van Hoecke,et al.  Neural Network Augmented Physics Models for Systems With Partially Unknown Dynamics: Application to Slider–Crank Mechanism , 2019, IEEE/ASME Transactions on Mechatronics.

[5]  C. Karen Liu,et al.  Data-Augmented Contact Model for Rigid Body Simulation , 2018, L4DC.

[6]  Satish Karra,et al.  AdjointNet: Constraining machine learning models with physics-based codes , 2021, ArXiv.

[7]  Ryan P. Adams,et al.  Amortized Synthesis of Constrained Configurations Using a Differentiable Surrogate , 2021, NeurIPS.

[8]  Mihaela van der Schaar,et al.  Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression , 2021, NeurIPS.

[9]  Hai V. Nguyen,et al.  Model-Constrained Deep Learning Approaches for Inverse Problems , 2021, ArXiv.

[10]  Sebastian Houben,et al.  Explainable Machine Learning with Prior Knowledge: An Overview , 2021, ArXiv.

[11]  U. T. Tygesen,et al.  Grey-box models for wave loading prediction , 2021, ArXiv.

[12]  Yigit A. Yucesan,et al.  Estimating model inadequacy in ordinary differential equations with physics-informed neural networks , 2021, Computers & Structures.

[13]  Daniel Cremers,et al.  Variational Data Assimilation with a Learned Inverse Observation Operator , 2021, ICML.

[14]  Yoshua Bengio,et al.  Towards Causal Representation Learning , 2021, ArXiv.

[15]  C. K. Liu,et al.  SimGAN: Hybrid Simulator Identification for Domain Adaptation via Adversarial Reinforcement Learning , 2021, 2021 IEEE International Conference on Robotics and Automation (ICRA).

[16]  Phaedon-Stelios Koutsourelakis,et al.  Physics-aware, probabilistic model order reduction with guaranteed stability , 2021, ICLR.

[17]  Simon W. Funke,et al.  Hybrid FEM-NN models: Combining artificial neural networks with the finite element method , 2021, J. Comput. Phys..

[18]  Burak Aksoylu,et al.  Physics guided machine learning using simplified theories , 2020, Physics of Fluids.

[19]  Rahul Rai,et al.  MIDPhyNet: Memorized infusion of decomposed physics in neural networks to model dynamic systems , 2020, Neurocomputing.

[20]  G. Sukhatme,et al.  NeuralSim: Augmenting Differentiable Simulators with Neural Networks , 2020, 2021 IEEE International Conference on Robotics and Automation (ICRA).

[21]  Emmanuel de B'ezenac,et al.  Augmenting physical models with deep networks for complex dynamics forecasting , 2020, ICLR.

[22]  Ekin D. Cubuk,et al.  Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics , 2020, Physical review letters.

[23]  J. Schneider,et al.  Neural Dynamical Systems: Balancing Structure and Flexibility in Physical Prediction , 2020, 2021 60th IEEE Conference on Decision and Control (CDC).

[24]  Phaedon-Stelios Koutsourelakis,et al.  A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables , 2020, J. Comput. Phys..

[25]  Liu Yang,et al.  B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data , 2020, J. Comput. Phys..

[26]  Uri Shalit,et al.  Generative ODE modeling with known unknowns , 2020, CHIL.

[27]  Pouria A. Mistani,et al.  Solving inverse-PDE problems with physics-aware neural networks , 2020, J. Comput. Phys..

[28]  Yonina C. Eldar,et al.  Model-Based Deep Learning , 2020, Proceedings of the IEEE.

[29]  Congjie Wei,et al.  Thermodynamic Consistent Neural Networks for Learning Material Interfacial Mechanics , 2020, ArXiv.

[30]  Chen Zeng,et al.  A physics-aware learning architecture with input transfer networks for predictive modeling , 2020, Appl. Soft Comput..

[31]  Luca Martino,et al.  Living in the Physics and Machine Learning Interplay for Earth Observation , 2020, ArXiv.

[32]  Carl Edward Rasmussen,et al.  Ensembling geophysical models with Bayesian Neural Networks , 2020, NeurIPS.

[33]  Long T. Le,et al.  Interpretable Sequence Learning for COVID-19 Forecasting , 2020, NeurIPS.

[34]  J. Zico Kolter,et al.  Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction , 2020, ICML.

[35]  Nils Thuerey,et al.  Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers , 2020, NeurIPS.

[36]  Theodoros Damoulas,et al.  Variational Autoencoding of PDE Inverse Problems , 2020, ArXiv.

[37]  Kira Radinsky,et al.  SimGANs: Simulator-Based Generative Adversarial Networks for ECG Synthesis to Improve Deep ECG Classification , 2020, ICML.

[38]  Thomas A. Runkler,et al.  Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics , 2020, IFAC-PapersOnLine.

[39]  Rafet Sifa,et al.  Combining Machine Learning and Simulation to a Hybrid Modelling Approach: Current and Future Directions , 2020, IDA.

[40]  John S. Baras,et al.  Interpretable machine learning models: a physics-based view , 2020, ArXiv.

[41]  Vipin Kumar,et al.  Integrating Physics-Based Modeling with Machine Learning: A Survey , 2020, ArXiv.

[42]  Miles Cranmer,et al.  Lagrangian Neural Networks , 2020, ICLR 2020.

[43]  Nicolas Thome,et al.  Disentangling Physical Dynamics From Unknown Factors for Unsupervised Video Prediction , 2020, 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[44]  Michael Chertkov,et al.  Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence , 2020, 2002.00021.

[45]  Ali Ramadhan,et al.  Universal Differential Equations for Scientific Machine Learning , 2020, ArXiv.

[46]  Phaedon-Stelios Koutsourelakis,et al.  Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems , 2019, J. Comput. Phys..

[47]  Gilles Louppe,et al.  The frontier of simulation-based inference , 2019, Proceedings of the National Academy of Sciences.

[48]  M. Deisenroth,et al.  Variational Integrator Networks for Physically Structured Embeddings , 2019, AISTATS.

[49]  Danilo Jimenez Rezende,et al.  Hamiltonian Generative Networks , 2019, ICLR.

[50]  Yang Liu,et al.  Physics-guided Convolutional Neural Network (PhyCNN) for Data-driven Seismic Response Modeling , 2019, Engineering Structures.

[51]  Miguel A. Aragon-Calvo Self-supervised Learning with Physics-aware Neural Networks I: Galaxy Model Fitting , 2020 .

[52]  Timothy M. Hospedales,et al.  Physics-as-Inverse-Graphics: Unsupervised Physical Parameter Estimation from Video , 2019, ICLR.

[53]  Alberto Rodriguez,et al.  TossingBot: Learning to Throw Arbitrary Objects With Residual Physics , 2019, IEEE Transactions on Robotics.

[54]  Physics-informed Generative Adversarial Networks for Sequence Generation with Limited Data , 2020 .

[55]  Nikhil Muralidhar,et al.  PhyNet: Physics Guided Neural Networks for Particle Drag Force Prediction in Assembly , 2020, SDM.

[56]  Fred Moolekamp,et al.  Hybrid Physical-Deep Learning Model for Astronomical Inverse Problems , 2019, ArXiv.

[57]  Ion Matei,et al.  PI-LSTM: Physics-Infused Long Short-Term Memory Network , 2019, 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA).

[58]  Ilaria Carpinella,et al.  Human kinematic, kinetic and EMG data during different walking and stair ascending and descending tasks , 2019, Scientific Data.

[59]  Qi Wang,et al.  Integrating Model-Driven and Data-Driven Methods for Power System Frequency Stability Assessment and Control , 2019, IEEE Transactions on Power Systems.

[60]  Achuta Kadambi,et al.  Blending Diverse Physical Priors with Neural Networks , 2019, ArXiv.

[61]  Stefano Ermon,et al.  InfoVAE: Balancing Learning and Inference in Variational Autoencoders , 2019, AAAI.

[62]  Jan Peters,et al.  Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning , 2019, ICLR.

[63]  Jason Yosinski,et al.  Hamiltonian Neural Networks , 2019, NeurIPS.

[64]  Jiajun Wu,et al.  Combining Physical Simulators and Object-Based Networks for Control , 2019, 2019 International Conference on Robotics and Automation (ICRA).

[65]  Joachim Denzler,et al.  Deep learning and process understanding for data-driven Earth system science , 2019, Nature.

[66]  Paris Perdikaris,et al.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , 2019, J. Comput. Phys..

[67]  Bin Dong,et al.  PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network , 2018, J. Comput. Phys..

[68]  Anuj Karpatne,et al.  Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles , 2018, SDM.

[69]  Alexandre M. Tartakovsky,et al.  Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks , 2018, J. Comput. Phys..

[70]  Markus Heinonen,et al.  ODE2VAE: Deep generative second order ODEs with Bayesian neural networks , 2019, NeurIPS.

[71]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[72]  Yibo Yang,et al.  Physics-informed deep generative models , 2018, ArXiv.

[73]  Luca Saglietti,et al.  Gaussian Process Prior Variational Autoencoders , 2018, NeurIPS.

[74]  Pierre-Yves Oudeyer,et al.  Sim-to-Real Transfer with Neural-Augmented Robot Simulation , 2018, CoRL.

[75]  Xinlei Chen,et al.  PGA: Physics Guided and Adaptive Approach for Mobile Fine-Grained Air Pollution Estimation , 2018, UbiComp/ISWC Adjunct.

[76]  Rishee K. Jain,et al.  Data-driven Urban Energy Simulation (DUE-S): A framework for integrating engineering simulation and machine learning methods in a multi-scale urban energy modeling workflow , 2018, Applied Energy.

[77]  Jo Bovy,et al.  Deep learning of multi-element abundances from high-resolution spectroscopic data , 2018, Monthly Notices of the Royal Astronomical Society.

[78]  Leslie Pack Kaelbling,et al.  Augmenting Physical Simulators with Stochastic Neural Networks: Case Study of Planar Pushing and Bouncing , 2018, 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[79]  Saibal Mukhopadhyay,et al.  HybridNet: Integrating Model-based and Data-driven Learning to Predict Evolution of Dynamical Systems , 2018, CoRL.

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

[81]  Stefano Ermon,et al.  Learning with Weak Supervision from Physics and Data-Driven Constraints , 2018, AI Mag..

[82]  Petros Koumoutsakos,et al.  Data-assisted reduced-order modeling of extreme events in complex dynamical systems , 2018, PloS one.

[83]  Stephan Mandt,et al.  Disentangled Sequential Autoencoder , 2018, ICML.

[84]  S. Ermon,et al.  The Information-Autoencoding Family: A Lagrangian Perspective on Latent Variable Generative Modeling , 2018 .

[85]  Maziar Raissi,et al.  Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations , 2018, J. Mach. Learn. Res..

[86]  Patrick Gallinari,et al.  Deep learning for physical processes: incorporating prior scientific knowledge , 2017, ICLR.

[87]  Bin Dong,et al.  PDE-Net: Learning PDEs from Data , 2017, ICML.

[88]  Anuj Karpatne,et al.  Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling , 2017, ArXiv.

[89]  Ming-Chang Wu,et al.  A physically based and machine learning hybrid approach for accurate rainfall-runoff modeling during extreme typhoon events , 2017, Appl. Soft Comput..

[90]  J. Zico Kolter,et al.  OptNet: Differentiable Optimization as a Layer in Neural Networks , 2017, ICML.

[91]  Jochen J. Steil,et al.  Hybrid Analytical and Data-Driven Modeling for Feed-Forward Robot Control † , 2017, Sensors.

[92]  Nagiza F. Samatova,et al.  Theory-Guided Data Science: A New Paradigm for Scientific Discovery from Data , 2016, IEEE Transactions on Knowledge and Data Engineering.

[93]  Christopher Burgess,et al.  beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework , 2016, ICLR 2016.

[94]  Uri Shalit,et al.  Structured Inference Networks for Nonlinear State Space Models , 2016, AAAI.

[95]  Stefano Ermon,et al.  Label-Free Supervision of Neural Networks with Physics and Domain Knowledge , 2016, AAAI.

[96]  Maximilian Karl,et al.  Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data , 2016, ICLR.

[97]  Ole Winther,et al.  Sequential Neural Models with Stochastic Layers , 2016, NIPS.

[98]  Yoshua Bengio,et al.  A Recurrent Latent Variable Model for Sequential Data , 2015, NIPS.

[99]  Garching,et al.  Imfit: A Fast, Flexible New Program for Astronomical Image Fitting , 2014, 1408.1097.

[100]  Daan Wierstra,et al.  Stochastic Backpropagation and Approximate Inference in Deep Generative Models , 2014, ICML.

[101]  Max Welling,et al.  Auto-Encoding Variational Bayes , 2013, ICLR.

[102]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[103]  Neil D. Lawrence,et al.  Latent Force Models , 2009, AISTATS.

[104]  D. Gordon E. Robertson,et al.  Research Methods in Biomechanics , 2004 .

[105]  I.G. Kevrekidis,et al.  Continuous-time nonlinear signal processing: a neural network based approach for gray box identification , 1994, Proceedings of IEEE Workshop on Neural Networks for Signal Processing.

[106]  Mark A. Kramer,et al.  Modeling chemical processes using prior knowledge and neural networks , 1994 .

[107]  Lyle H. Ungar,et al.  A hybrid neural network‐first principles approach to process modeling , 1992 .