Unsupervised Learning Of Latent Physical Properties Using . - UAI
Unsupervised Learning of Latent Physical Properties UsingPerception-Prediction NetworksDavid Zheng1 , Vinson Luo2 , Jiajun Wu1 , and Joshua B. Tenenbaum11Computer Science and Artificial Intelligence Laboratory, MIT2Department of Computer Science, Stanford UniversityAbstractWe propose a framework for the completelyunsupervised learning of latent object properties from their interactions: the perceptionprediction network (PPN). Consisting of a perception module that extracts representations oflatent object properties and a prediction modulethat uses those extracted properties to simulatesystem dynamics, the PPN can be trained inan end-to-end fashion purely from samples ofobject dynamics. The representations of latentobject properties learned by PPNs not only aresufficient to accurately simulate the dynamicsof systems comprised of previously unseen objects, but also can be translated directly intohuman-interpretable properties (e.g. mass, coefficient of restitution) in an entirely unsupervised manner. Crucially, PPNs also generalizeto novel scenarios: their gradient-based trainingcan be applied to many dynamical systems andtheir graph-based structure functions over systems comprised of different numbers of objects.Our results demonstrate the efficacy of graphbased neural architectures in object-centric inference and prediction tasks, and our model hasthe potential to discover relevant object properties in systems that are not yet well understood.1INTRODUCTIONThe physical properties of objects, combined with thelaws of physics, govern the way in which objects moveand interact in our world. Assigning properties to objects we observe helps us summarize our understandingof those objects and make better predictions of their futurebehavior. Often, the discovery of such properties can beperformed with little supervision. For instance, by watching an archer shoot several arrows, we may concludethat properties such as the tension of the bowstring, thestrength and direction of the wind, and the mass and dragcoefficient of the arrow affect the arrow’s ultimate trajectory. Even when given observations from entirely novelmicroworlds, humans are still able to learn the relevantphysical properties that characterize a system [1].Our work utilizes recent advances in neural relation networks in order to learn latent physical properties of asystem in an unsupervised manner. In particular, the neural relation architectures [2, 3] have proven capable ofaccurately simulating complex physical interactions involving objects with known physical properties. Relationnetworks have several characteristics that make them particularly suitable for our task: they are fully differentiable,allowing them to be applied to a variety of different situations without the need for any architectural change; theyhave a modular graph-based structure that generalizesover differing numbers of objects; and their basic architecture can be easily applied to both dynamics predictionand the learning of latent properties.We use relation networks to construct the perceptionprediction network (PPN), a novel system that uses arepresentation learning [4] paradigm to extract an encoding of the properties of a physical system purely throughobservation. Unlike previous neural relation architectures,which only use relation networks to predict object stateswith known property values, we use relation networks tocreate both a perception network, which derives propertyvalues from observations, and a prediction network, whichpredicts object positions given property values. The PPNis able to derive unsupervised representations of the latent properties relevant to physical simulations purely byobserving the dynamics of systems comprised of objectswith different property values. These learned representations can be translated directly into human-interpretableproperties such as mass and coefficient of restitution.One crucial aspect of our system is generalization, whichhumans excel at when inferring latent properties of novel
Trajectory forObject SetNew Trajectory forObject SetProperty VectorProperty VectorHuman-interpretablePropertiesMassProperty VectorPerceptionUnsupervisedAlgorithmsPrediction Coefficient ofRestitutionChargeFigure 1: Model overview. The unsupervised object property discovery paradigm that the PPN follows extractsproperty vectors from samples of object dynamics to accurately predict new trajectories of those same objects. Applyingunsupervised learning methods to the learned vectors allows for the extraction of human-interpretable object properties.systems. Our proposed system is robust under severalforms of generalization, and we present experimentsdemonstrating the ability of our unsupervised approachto discern interpretable properties even when faced withdifferent numbers of objects during training and testingas well as property values in previously unseen ranges.We evaluate the PPN for two major functionalities: theaccuracy of dynamics prediction for unseen objects andthe interpretability of properties learned by the model. Weshow that our model is capable of accurately simulatingthe dynamics of complex multi-interaction systems withunknown property values after only a short observationalperiod to infer those property values. Furthermore, wedemonstrate that the representations learned by our modelcan be easily translated into relevant human-interpretableproperties using entirely unsupervised methods. Additionally, we use several experiments to show that both theaccuracy of dynamics prediction and interpretability ofproperties generalize well to new scenarios with differentnumbers and configurations of objects. Ultimately, thePPN serves as a powerful and general framework for discovering underlying properties of a physical system andsimulating its dynamics.2RELATED WORKPrevious methods of modeling intuitive physics havelargely fallen under two broad categories: top-down approaches, which infer physical parameters for an existingsymbolic physics engine [1, 5, 6, 7, 8, 9], and bottomup approaches, which directly predict physical quantitiesor future motion given observations [10, 11, 12, 13, 14,15, 16]. While top-down approaches are able to generalize well to any situation supported by their underlyingphysics engines (e.g. different numbers of objects, previously unseen property values, etc.), they are difficultto adapt to situations not supported by their underlyingdescription languages, requiring manual modificationsto support new types of interactions. On the other hand,bottom-up approaches are often capable of learning thedynamics of formerly unseen situations without any fur-ther modification, though they often lack the ability togeneralize in the same manner as top-down approaches.Recently, a hybrid approach has used neural relation networks, a specific instance of the more general class ofgraph-based neural networks [17, 18], to attain the generalization benefits of top-down approaches without requiring an underlying physics engine. Relation networksrely on the use of a commutative and associative operation (usually vector addition) to combine pairwise interactions between object state vectors in order to predictfuture object states [19]. These networks have demonstrated success in simulating multiple object dynamicsunder interactions including Coulomb charge, object collision (with and without perfect elasticity), and springtension [2, 3, 20, 21]. Much like a top-down approach,relation networks are able to generalize their predictionsof object position and velocity to different numbers of objects (training on 6 objects and testing on 9, for instance)without any modification to the network weights; furthermore, they are fully differentiable architectures that canbe trained via gradient descent on a variety of interactions.Our paper leverages the interaction network in a novelway, demonstrating for the first time its efficacy as a perception module and as a building block for unsupervisedrepresentation learning.Additional research has looked at the supervised and unsupervised learning of latent object properties, attemptingto mirror the inference of object properties that humansare able to perform in physical environments [1]. Wu etal. [9] leverages a deep model alongside set physical lawsto estimate properties such as mass, volume, and materialfrom raw video input. Fraccaro et al. [22] uses a variational autoencoder to derive the latent state of a singlebouncing ball domain, which they then simulate usingKalman filtering. Chang et al. [3] demonstrate that theirrelation network based physics simulator is also capable of performing maximum-likelihood inference over adiscrete set of possible property values by comparing simulation output for each possibility to reality. Our papergoes one step further by showing that physical proper-
Perception NetworkPrediction Network Input: Observed States, Initial Rollout StateOutput: Predicted StatesFigure 2: Model architecture. The PPN takes as input a sequence of observed states O1 , . . . , OT as well an initial stateR0 to begin a new rollout. Code vectors C1 , . . . , CT are derived from the observed states using interaction networks anda final property vector Z is produced by the perception network. The property vector is then utilized by the predictionnetwork to recursively predict future object states R1 , R2 , . . . for a new rollout given initial state R0 . We train the PPNto minimize the L2 distance between the predicted rollout states and the ground truth states for those timesteps.ties can be learned from no more than raw motion dataof multiple objects. Recently, Kipf et al. [23] has alsoutilized relation networks to infer the identity of categorical interactions between objects; in contrast, our paper isconcerned with the learning of object properties.3MODEL3.1PERCEPTION-PREDICTION NETWORKThe PPN observes the physical dynamics of objects withunknown latent properties (e.g. mass, coefficient of restitution) and learns to generate meaningful representationsof these object properties that can be used for later simulations. An overview of the full network is shown in Figure1. The PPN consists of the following two components: The perception network takes as input a sequenceof frames on the movements of objects over a shortobservation window. It outputs a property vector foreach object in the scene that encodes relevant latentphysical properties for that object. Each input frameis a set of state vectors, consisting of each object’sposition and instantaneous velocity. During training,no direct supervision target is given for the propertyvectors. The prediction network uses the property vectorsgenerated by the perception network to simulate theobjects from a different starting configuration. Thenetwork takes as input the property vectors generated by the perception network and new initial statevectors for all objects. Its output is a rollout of theobjects’ future states from their new starting state.The training target for the prediction network is theground truth states of the rollout sequence.We implement both the perception and prediction networks using interaction networks [2], a specific type ofneural relation network that is fully differentiable andgeneralizes to arbitrary numbers of objects. This enablesus to train both networks end-to-end using gradient descent with just the supervision signal of the predictionnetwork’s rollout target, as the property vectors output bythe perception network feed directly into the predictionnetwork.3.2INTERACTION NETWORKAn interaction network (IN) is a relation network thatserves as the building block for both the perception andprediction networks. At a high level, interaction networks use multilayer perceptrons (MLPs) to implementtwo modular functions, the relational model frel and theobject model fobj , which are used to transform a set ofobject-specific input features {x(1) , . . . , x(N ) } into a setof object-specific output features {y (1) , . . . , y (N ) }, whereN is the number of objects in a system. Given input features for two objects i and j, frel calculates the “effect”vector of object j on object i as e(i,j) frel (x(i) , x(j) ).The net effecti, e(i) , is the vector sum of all pairPon object(i,j)wise effects j6 i eon object i. Finally, the output forobject i is given by y (i) fobj (x(i) , e(i) ). Importantly,fobj and frel are shared functions that are applied overall objects and object-object interactions, allowing thenetwork to generalize across variable numbers of objects.Interaction networks are capable of learning state-to-statetransition functions for systems with complex physicaldynamics. More generally, however, interaction networkscan be used to model functions where input and outputfeatures are specific to particular objects and the relationship between input and output is the same for each object.
While our prediction network uses an interaction networkto simulate state transitions, our perception network usesan interaction network to make incremental updates on thevalues of object latent properties from observed evidence.3.3PERCEPTION NETWORKThe perception network produces object-specific propertyvectors, Z, from a sequence of observed states O. Asshown in Figure 2, our perception network is a recurrentneural network that uses an interaction network as its corerecurrent unit. The perception network begins with objectspecific code vectors, C1 , initialized to zero vectors, withsome fixed size LC for each object. At each step t, the INtakes in the previous code vectors, Ct 1 , as well as thelast two observed states, Ot 1 and Ot , to produce updatedcode vectors, Ct , also of size LC . After processing all TOobservation frames, the perception network feeds the finalcode vectors CTO into a single code-to-property MLP thatconverts each object’s code vector into an “uncentered”property vector of size LZ per object. We denote the finalcollection of uncentered property vectors as Zu .In many physical systems, it may be impossible or undesirable to measure the latent properties of objects onan absolute scale. For example, in a system where twoballs collide elastically, a collision can only inform uson the mass of each object relative to the other object,not their absolute mass values. In order to allow for theinference of absolute property values, we let the first object of every system serve as a reference object and takeon the same property values in each system. In doingso, we can infer the absolute property values of all otherobjects by observing their value relative to the referenceobject. To enforce inference relative to the reference object, we “center” the property vectors by subtracting thereference object’s uncentered property vector from eachobject’s uncentered property vector, producing the finalproperty vectors Z. Note that this ensures that the reference object’s property vector is always a zero vector,agreeing with the fact that its properties are known to beconstant. We can summarize the perception network withthe following formulas:C1 0(1)Ct INpe (Ct 1 kOt 1 kOt ), for t 2, . . . , TO (2) (i)Zu(i) MLPpe CTO , for i 1, . . . , N(3)Z(i) Zu(i) Zu(1) ,for i 1, . . . , N(4)where k is the object-wise concatenation operator, INpe isthe perception interaction network, MLPpe is the code-to(1)property MLP, and Zu is the reference object’s uncentered property vector.3.4PREDICTION NETWORKThe prediction network performs state-to-state rollouts ofthe system from a new initial state, R0 , using the propertyvectors produced by the perception network. Like theperception network, the prediction network is a recurrentneural network with an Interaction Network core. At stept, the IN takes in the previous state vectors, Rt 1 , and theproperty vectors, Z, and outputs a prediction of the nextstate vectors, Rt . In other words,Rt INpr (Rt 1 kZ), for t 1, ., TR(5)where INpr is the prediction interaction network and TRis the number of rollout frames.The prediction loss for the model is the total MSE betweenthe predicted and true values of {Rt }t 1.TR .4EXPERIMENTS4.1PHYSICAL SYSTEMSFor our experiments, we focus on 2-D domains whereboth the latent property inference task and the subsequentdynamics prediction task are challenging. In all systems,the first object serves as the reference object and has fixedproperties. All other objects’ properties can be inferredrelative to the reference object’s properties. We evaluatethe PPN on the following domains (see Fig. 5): Springs Balls of equal mass have a fictitious property called “spring charge” and interact as if all pairsof objects were connected by springs governed byHooke’s law . The reference object has a springcharge of 1, while all other objects have springcharges selected independently at random from thelog-uniform† distribution over [0.25, 4]. The springconstant of the spring connecting any given pair ofobjects is the product of the spring charges of the twoobjects, and the equilibrium distance for all springsis a fixed constant. Perfectly Elastic Bouncing Balls Balls of fixed radius bounce off each other elastically in a closed box.The reference object has a mass of 1. Each other ballhas a mass selected independently at random fromthe log-uniform distribution over [0.25, 4]. The fourwalls surrounding the balls have infinite mass anddo not move. Two objects connected by a spring governed by Hooke’slaw are subject to a force F k(x x0 ), where k is thespring constant of the spring, x is the distance between the twoobjects, and x0 is the spring’s equilibrium distance. The forceis directed along the line connecting the two objects but variesin sign: it is attractive if x x0 and repulsive if x x0 .†We use the phrase log-uniform distribution over [A, B] toindicate the distribution of exp(x), where x is drawn uniformlyat random over the interval [log A, log B].
Inelastic Bouncing Balls Building off the previous domain, we introduce additional complexity byadding coefficient of restitution (COR) as anothervarying latent property of each object. The CORof a collision is the ratio of the final to initial relative velocity between the two colliding objects alongthe axis perpendicular to the contact plane. In aperfectly elastic domain, for example, all collisionswould have a COR of 1. In our new domain, eachobject has a random COR selected uniformly from[0.5, 1]. The reference object has a COR of 0.75.The COR used to compute the dynamics in a collision between two balls is defined as the maximumof the two colliding objects’ CORs. When a ballcollides with a wall, the ball’s COR is used for thecollision.For each domain, we train the PPN on a 6-object datasetwith 106 samples and validate on a 6-object dataset with105 samples. Each sample consists of 50 observationframes used as input into the perception network and 24rollout frames used as targets by the prediction network.We evaluated our model on 3-object, 6-object, and 9object test sets, each with 105 samples.In addition, we also wish to demonstrate the PPN’s abilityto generalize to new objects whose latent properties areoutside of the range of values seen during training. Forthis experiment, we test our model on a new 2-objectperfectly elastic balls dataset with 105 samples. The massof the first ball remains fixed at 1, while the mass of thesecond ball is selected from 11 values ranging from 32 1to 32, spaced evenly on a log scale. We perform a similarexperiment on the springs domain, using the same 11values as the spring charge of the second object.We use matter-js‡ , a general-purpose rigid-body physicsengine, to generate ground truth data. In all simulations,balls are contained in a 512 px 512 px closed box. Eachball has a 50 px radius and randomly initialized positionssuch that no ball overlaps. In the springs domain, initialx- and y-velocity components are selected uniformly atrandom from the range [ 15, 15] px/sec, the equilibriumdisplacement for each spring is 150, and the mass of allballs is 104 . In the perfectly elastic balls domain, initialvelocity components are selected from the range [ 9, 9]px/sec. In the inelastic balls domain, they are selectedfrom the range [ 13, 13] px/sec. Each dataset’s framesare sampled at 120 fps.In the creation of our bouncing ball datasets, we use rejection sampling to filter out simulations in which someobject latent properties cannot be inferred from the observation frames. In both bouncing ball domains, we must‡http://brm.io/matter-js/be able to infer the mass of every object. In order toguarantee this, each object must collide directly with thereference object or be linked indirectly to it through asequence of collisions. For the inelastic domain, we mustensure that each object’s COR can be inferred as well. Ina ball-ball collision, only the higher object COR is usedin determining collision dynamics, and so only the higherobject COR can be inferred from the collision. For thisreason, every ball must either collide with a ball of lowerCOR or a wall.4.2MODEL ARCHITECTUREWe use a single model architecture for all of our experiments. We set LC , the size of each code vector, to 25and LZ , the size of each property vector, to 15. All MLPsin the model, including those in the interaction networks,use linear hidden layers with ReLU activation and a linearoutput layer.Following the overall structure of Battaglia et al. [2],the perception network’s IN core consists of a 4-layerrelation-centric MLP with sizes [75, 75, 75, 50] and a 3layer object-centric MLP with sizes [50, 50, 25]. The finalcode vectors output by the IN feed into another objectcentric MLP of size [15, 15, 15] to produce the final latentproperty vectors of size 15. The prediction network’s INcore consists of a 5-layer relation-centric MLP with sizes[100, 100, 100, 100, 50] and a 3-layer object-centric MLPwith sizes [50, 50, 4] used to predict each object’s nextposition and velocity.The perception network and prediction network aretrained end-to-end using a single training loss, whichwe call the prediction loss. The prediction loss is theunweighted sum of the MSE of the predicted vs actualstate vectors of all objects during the 24 rollout timesteps.In addition, we apply L2 regularization on the “effects”layer of both the perception and prediction networks. Thisregularization encourages minimal information exchangeduring interactions and proves to be a crucial componentto generalization to different numbers of objects. We selected the penalty factor for each regularization term viagrid search. We also experimented with the use of β-VAEregularization [24, 25] on property vectors to encouragethe learning of interpretable and factorized properties.In order to improve stability when simulating long rollouts, we added a small amount of Gaussian noise toeach state vector during rollout, forcing the model toself-correct for errors. Empirically, we found that settingthe noise std. dev. equal to 0.001 the std. dev. of eachstate vector element’s values across the dataset stabilizedrollout positions without affecting loss.We trained the model for 150 epochs and optimized theparameters using Adam [26] with mini-batch size 256.
SpringsPerfectly Elastic BallsInelastic BallsComponent #EVRR2 w/ log chargeEVRR2 w/ log .730.270.0060R2 w/ log mass R2 w/ COR0.900.02000.020.8100Table 1: Principal component analysis. Applying PCA on the property vectors yields principal components that arehighly correlated with human-interpretable latent properties such as COR and the log of mass. We compute statistics onthe first four principal components of the property vectors for each training set. Explained variance ratio or EVR isthe explained variance of the principal component as a fraction of overall variance, and R2 is the squared in-samplecorrelation between the principal component and a particular ground truth property. Values less than 10 3 round to 0.# Training Data# Test ObjectsSprings2Perfectly Elastic Balls2Inelastic Balls2R w/ log chargeR w/ log massR w/ log massR2 w/ COR1052 1055 .900.860.68Table 2: Data-efficiency and number of objects generalization. The PPN learns to capture physical properties with105 training data points and converges when given 2 105 instances. Its predictions generalize well to out-of-sampletest sets with varying numbers of objects. We train the PPN on a 6-object dataset and test it on entirely new datasetscomprised of 6, 3, and 9 objects. Above, we report the R2 when using the property vector’s first principal component topredict log mass and the second principal component to predict COR (for the inelastic balls case). Note that even in the3 and 9 object cases the PPN is able to extract mass and coefficient of restitution with high R2 .We used a waterfall schedule that began with a learningrate of 5 10 4 and downscaled by 0.8 each time thevalidation error, estimated over a window of 10 epochs,stopped decreasing.55.1RESULTSEXTRACTING LATENT PROPERTIESOur results show that the physical properties of objectsare successfully encoded in the property vectors outputby the perception network. In fact, we can extract thehuman-interpretable notions of spring charge, mass, andCOR by applying principal component analysis (PCA) tothe property vectors generated by the perception networkduring training. We find that the first principal componentof each property vector is highly correlated with the log ofspring charge in the spring domain and the log of objectmass in both bouncing ball domains. In the inelastic ballsdomain, we also find that the second principal componentof the property vector is highly correlated with COR. Table 1 shows the explained variance ratio (EVR) of eachof the first 4 principal components of the learned propertyvectors in all three domains, along with the R2 when eachcomponent is used to predict ground truth object prop-erties§ . Since PCA is an unsupervised technique, thesescalar quantities can be discovered without prior notionsof mass and COR, and we can use the order-of-magnitudedifference between certain principal components’ EVRto identify which components represent meaningful properties and which merely capture noise.We also find that each learned property vector only contains information about its associated object and not anyother objects. We test this hypothesis by using linear leastsquares to calculate the in-sample R2 between the groundtruth latent properties of each object and the concatenation of the property vectors of all other objects. This R2is less than 5% for each of the three domains and theirrelevant latent properties.In order to test the generalization properties of our perception network, we calculate the out-of-sample R2 whenusing the perception network (trained on 6 object dynamics) and PCA to predict property values for test sets withvarying number of objects, as shown in Table 2. The table§By default, the property values produced by PCA will notbe in the same scale as our ground truth values. For the purposesof correlation analysis, we linearly scale predictions to matchthe mean and std. dev. of the ground truth latent values.
Inelastic Balls1.00.80.8OOS R2OOS R2Perfectly Elastic Balls1.00.60.40.20.00.60.40.20123450.001Reference Distance2345Reference DistanceFigure 3: Mass prediction vs. reference distance. Out-of-sample R2 on the two 6-object bouncing balls datasets forpredicting log mass at different reference distances. The PPN must combine a sequence of intermediate mass inferencesto accurately infer the mass of an object with large reference distance.also shows how PPN performs when given a differentnumber of training instances. In all bouncing balls testsets, for our model trained on 106 data points, the OOSR2 for log mass is above 90%, the OOS R2 for COR isabove 68%, and the OOS R2 for log spring charge in thesprings domain is above 87%.We also compare the PPN against a LSTM-PPN baseline. The LSTM-PPN replaces each of the perceptionand prediction networks in the PPN with stacked LSTMs.Unlike an interaction network, an LSTM does not factorize input and output by object. Instead, state vectors foreach object are concatenated and processed together, anda single property vector is learned for all objects. Table3 shows that the LSTM-PPN does not learn meaningfullatent properties. In each scenario, the linear least squaresin-sample R2 between true object properties and propertyvectors is less than 2%. We also experiment with differentvalues of β in the regularization term of the property vectors Z as in β-VAE [25]. The value of β does not impactthe PPN’s performance on learning object properties.For the two bouncing balls domains, the relative masses ofobjects are inferred through collisions, but not all objectscollide directly with the reference object. We define thereference distance of an object to be the minimum numberof collisions needed during observation to relate the object’s mass to that of the reference object. Inference on anobject with reference distance of 3, for example, dependson the inference of the mass of two intermediate objects.Figure 3 shows the relation between the PPN’s predictionR2 and reference distance for each of the 6-object testsets. While there is a decay in R2 as reference distanceincreases due to compounding errors during inference,the PPN clearly demonstrates the ability to use transitivityto infer the mass of objects with large reference distance.5.2ROLLOUT PREDICTIONSAlthough the PPN’s primary objective is the unsupervisedlearning of latent physical properties, the network canMethodsLSTMPPN (β 0)PPN (β 0.01)PPN (β 1)SpringsElastic Balls Inelastic Ballslog chargelog mass0.020.950.950.920.030.940.930.94log mass COR0.020.900.930.930.030.800.790.65Table 3: Comparing with baseline m
tent properties relevant to physical simulations purely by observing the dynamics of systems comprised of objects with different property values. These learned representa-tions can be translated directly into human-interpretable properties such as mass and coefficient of restitution. One crucial aspect of our system is generalization, which
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