Amortized Synthesis of Constrained Configurations Using a Differentiable Surrogate

1 Introduction . One of the ambitions of artificial intelligence is to automate problems in design , fabrication , and control that demand efficient and accurate interfaces between machine learning algorithms and physical systems . Whether it is optimizing the topology of a mechanical structure or identifying the feasible paths for a manufacturing robot , we can often view these problems through the lens of synthesis . In a synthesis task , we seek configurations of a physical system that achieve certain desiderata while satisfying given constraints ; i.e. , we must optimize a physically-realizable design . In this work , design refers to the parametric space over which we have control and in which , e.g. , we optimize . A realization is the object that arises when the design is instantiated , while goal refers to its desired properties . For example , in fabrication , the design might be a set of assembly steps , the realization would be the resulting object , while the goal could be to match target dimensions while maximizing strength . Synthesis , then , refers to finding a design whose realization achieves the goal . Synthesis problems are challenging for several reasons . The physical realization process may be costly and time-consuming , making evaluation of many designs difficult . Moreover , the realization process—or even a simulation of it—is generally not differentiable , rendering efficient gradient-based methods inapplicable . Finally , there may be a many-to-one map from the parametric space of feasible and equally-desirable designs to realizations ; i.e. , there may be multiple ways to achieve the goal . Surrogate modeling is widely used to address the first two challenges , though it can still be expensive because of the need for optimization , sampling , or search algorithms to find a feasible design . More seriously , the third challenge—lack of uniqueness—creates difficulties for naïve supervised learning 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . approaches to synthesis . Specifically , consider generating many design/realization pairs , evaluating the constraints and objectives on the realizations , and attempting to learn a supervised map from goal back to design . When multiple designs lead to the same realization , or multiple realizations achieve the same goal , the supervised learner is penalized for producing designs that are valid but happen to not be the ones used to generate the data . Moreover , this approach may learn to produce an “ average ” design that is actually incorrect . Figure 1a shows a cartoon example : if the goal is to throw a ball to reach some target distance , there are two possible launch angles ( designs ) resulting in landing points ( realizations ) at the correct spot . Performing least-squares regression from distance to angle on the full set of distance/angle pairs , however , learns an average angle that does not satisfy the goal . To address these challenges , we propose to use a two-stage neural network architecture that resembles an autoencoder . One stage ( the decoder ) acts as a differentiable surrogate capturing the many-to-one physical realization process . The other stage ( the encoder ) maps from a goal back to a design but , critically , it is trained end-to-end with a loss in the space of realizations that flows back through the decoder . Thus the encoder—our central object of interest for synthesis—is not constrained to match a specific design in a training dataset , but instead is tasked with finding any design that meets the desiderata of the realized output . The result is a neural network that performs amortized synthesis : it is trained once , and at run time produces a design that is approximately optimal , using only a feed-forward architecture . Note that our method is not an autoencoder , as the design is not a lower-dimensional representation of the goal , and the encoder and the decoder are trained in separate stages . Our method places a number of requirements on the synthesis problem . First , to train the surrogate , we need data pairs of designs and realizations . Commonly , this would require us to generate designs , in which a substantial amount of designs are viable , and simulate them on a simulator . Second , given our current setting , the physical realization process needs to be deterministic . Third , to train the encoder , the synthesis problem should have a clear objective function , or at least we can quantify the objective . Finally , given our current feed-forward setting , we consider synthesis problems that need only one valid design , although we may further extend our method by using a generative encoder . In this work , we demonstrate this two-stage approach on a pair of specific design tasks . The first case study is extruder path planning for a class of 3D printers ( the Markforged Mark Two ) that can reinforce polymer layers with discrete fibers ( Figure 1b ) . Since the fibers are stiff , their shape is deformed after extrusion ( Figure 1c , top row ) , and our task is to find an extruder path that results in a given fiber shape . As shown in Figure 1c , this problem has the many-to-one nature described above : for a small error tolerance on fiber path , there exist infinitely many extruder paths , which may even look very different . The second case study is constrained soft robot inverse kinematics . In this work , we use a simulation of a snake-like soft robot as in Xue et al . [ 97 ] , in which we can control the stretch ratios of each individual segment on both sides of the robot . The robot has to reach a target while avoiding an obstacle , and the locations of both are input goals . As before , there may be multiple solutions for a given goal ( i.e. , locations of target and obstacle ) , as shown in Figure 1d . For both case studies , we compare to two baseline algorithms . In direct-learning , a neural network for the synthesis problem ( i.e. , from goal to design ) is trained in a supervised manner on a set of designs . Since this effectively averages designs in the training dataset , as argued above , our method outperforms it significantly . The second baseline is direct-optimization , which uses a gradient-based method ( BFGS ) to optimize for each new design separately , given access to the trained differentiable surrogate for the realization process ( decoder ) . Our method is competitive with this rough “ performance upper bound ” while using dramatically lower computational resources . 2 Related work . Machine learning applications in synthesis problems . In synthesis tasks , the aim is to find a design solution such that its realization achieves one or more given goals . Usually , the solution is nonunique , and only one is needed . In molecule discovery , one would like to find a molecule which has some desired properties ( e.g. , minimal side effects , efficacy , metabolic stability , growth inhibition ) [ 20 , 46 , 71 , 41 , 83 ] . See Vamathevan et al . [ 90 ] , Chen et al . [ 16 ] for surveys on machine/deep learning in drug discovery . Challenges of molecule discovery include discrete design space [ 32 , 76 ] and limited data [ 47 ] due to difficulty in simulation . In materials synthesis , researchers seek materials with specific properties . See Bhuvaneswari et al . [ 7 ] for a review . Similarly , to obtain enough training data , researchers have put efforts into parsing and learning from scientific literature in natural language [ 52 , 45 ] . In 3D shape generation , one wants to find a 3D shape that has some desired properties : properties like 2.5D sketches [ 95 ] that can be easily calculated and properties like functionality [ 34 ] that need to be human-labeled . In topology optimization , researchers aim to maximize the system ’ s performance by optimizing the material layout given boundary conditions , constraints , external loads , etc [ 80 , 5 , 93 ] . Machine learning has been used to infer properties [ 75 ] , find representations of designs [ 48 , 99 ] , and directly generate designs [ 55 ] . In program synthesis , the goal is to find programs that realize given intentions ( e.g. , generating 3D shapes , answering visual questions ) [ 35 , 88 , 98 ] , where machine learning has been used to generate programs and execute programs . In this work , we propose an approach to learn a feed-forward neural network that can directly and efficiently produce a feasible solution to a synthesis problem that satisfies the requirements mentioned above . Surrogate/oracle-based synthesis . Surrogate/oracle-based synthesis uses an auxiliary model— a surrogate ( or sometimes called oracle ) —that can evaluate qualities of a design without timeconsuming laboratory experiments , while still being reasonably accurate [ 58 , 9 ] . Surrogate models can be physics-based or approximation-based [ 57 ] ( i.e. , empirical ) , and there are different modeling techniques for approximation-based surrogates , including polynomial regression , radial basis functions , Gaussian processes , and neural networks [ 58 , 39 ] . Bhosekar and Ierapetritou [ 6 ] provide a review of surrogate-based methods , and see Koziel and Leifsson [ 56 ] for a general review of surrogate models in engineering . There are two major approaches : optimization and sampling . The most common approach is surrogate-based optimization ; see Forrester and Keane [ 29 ] for a survey . Some application examples include : optimizing the parameters of a CPU simulator [ 72 ] ; solving partial differential equations in service of PDE-constrained optimization [ 97 ] ; and optimizing stochastic non-differentiable simulators [ 79 ] . Researchers have also developed methods to deal with the challenge that inputs can be out-of-distribution for the oracle [ 27 , 30 , 89 ] . On the other hand , sampling-based methods have several advantages : the design space can be discrete and can generate multiple designs [ 9 , 10 , 24 ] . Researchers have successfully applied sampling methods to problems in chemistry [ 32 ] and biology [ 36 , 51 , 73 ] . In this work , we use surrogate-based optimization as one of our baseline algorithms , and our proposed method uses a surrogate during training to optimize for a feed-forward network for the design problem . Differentiable surrogate of losses in machine learning . Since some loss functions in machine learning are not differentiable ( e.g. , IoU for rotated bounding boxes , 0-1 loss in classification ) , researchers have proposed to learn surrogates for them . Grabocka et al . [ 33 ] provided a formulation of surrogate loss learning and compared several learning mechanisms on some commonly used non-differentiable loss functions . Liu et al . [ 62 ] proposed a general pipeline to learn surrogate losses . Bao et al . [ 2 ] , Hanneke et al . [ 40 ] provided theoretical analyses of surrogates for 0-1 loss in classification . Patel et al . [ 70 ] , Nagendar et al . [ 64 ] , Yuan et al . [ 100 ] explored the use of surrogate losses in various real-world tasks , including medical image classification , semantic segmentation , and text detection and recognition . In this work , due to the non-uniqueness of design solutions , we further extend this idea from loss functions to more complex , physical realization processes . Robot motion planning . Robot motion planning [ 59 ] can also be viewed as a form of synthesis , and there are different types of approaches to it . One popular strategy is optimization . For example , researchers have used evolutionary algorithms ( EA ) [ 60 , 12 , 42 ] , including variants of genetic algorithms ( GA ) [ 18 , 17 , 87 , 11 , 50 , 53 , 28 ] and covariance matrix adaptation evolution strategy ( CMA-ES ) [ 37 ] . Another example is constrained optimization [ 94 , 85 , 21 , 23 ] , which includes extensions like sequential quadratic programming ( SQP ) [ 15 , 37 ] and the DIRECT algorithm [ 91 ] . Other examples of optimization applied to robot motion planning include reinforcement learning [ 81 , 25 ] and teaching-learning-based optimization [ 82 ] . Another popular class of methods is sampling , including simulated annealing ( SA ) [ 4 , 69 , 101 ] , probabilistic roadmaps ( PRM ) [ 49 ] , rapidly exploring random trees ( RRT ) [ 14 , 4 , 13 ] , etc . Finally , search methods , e.g. , A * search [ 38 ] , have been used to solve motion planning problems . These works usually model the robot ’ s physics directly without using a surrogate model and solve the design using methods including optimization , sampling , and search , which can be time-consuming . Our method amortizes the cost of inference , and can be used to solve planning problems with more complex physics that can not be directly modeled . In our first case study , we plan for a trajectory of the extruder , and our design space is not the speeds of the motors but rather the coordinates of points along the trajectory . In the second case study , rather than solving a dynamic planning problem , we solve a static planning task on a soft robot , with the stretch ratios of all the controllable segments of the robot as the design space . Path planning in 3D printing . Path planning is one of the most important problems in 3D printing , and people plan for different objectives : minimizing printing time , avoiding collision , faster planning , etc . Shembekar et al . [ 78 ] proposed a planning algorithm to build complex shapes with multiple curvatures and can avoid collisions . Ganganath et al . [ 31 ] minimized the printing time by modeling the task as a traveling salesman problem . Xiao et al . [ 96 ] speeded up path planning algorithms by introducing efficient topology reconstruction algorithms . Stragiotti [ 84 ] provided an optimizationbased algorithm that minimizes compliance of a printed part . Asif [ 1 ] introduced a planning algorithm for continuous fiber and can generate a continuous deposition path . See Huang et al . [ 43 ] for a review of existing work in 3D printing path design . In this work , instead of the aforementioned objectives , we plan an extruder path to compensate for the deformation caused by the printing process . We demonstrate increased run-time efficiency by amortizing the cost of physical simulation of the printing process , learning a feed-forward network to output the extruder path .

Summary: The paper proposes a novel method for addressing synthesis problems, where they define a synthesis as an object or configuration that meets a set of constraints while optimizing one or more objectives. The authors motivate the problem with manufacturing settings, such as an additive printing path and extruding sequence, as well as soft robotic motion problem that can be described by a partial differential equation (PDE). The authors aims to address a couple of challenges: * Provide a differentiable surrogate for non-differentiable simulations in synthesis problem. The claim and show that this can help with learning a differentiable model of the process that can help learn how to generate viable design via a synthesis process. * Propose and develop a method that can solve many-to-one synthesis problems using a two-stage where one neural network based model learns how to generate designs and another model learns how to evaluate designs. Since both models are differentiable, the authors can train their framework end-to-end, which they claim provides accuracy improvements compared with directly learning a design to evaluation map and compute improvement compared to direct optimization of design to evaluation using method like BFGS. * Provide a series of objective functions that can evaluate various synthesis design for a diverse set of use cases, which the authors then use to inform their two-stage approach.  The authors outline the objective function for an extruder path planning case and a constrained soft robotic motion task. In both cases they optimize the objective and perform the synthesis task using their approach as well as the direct learning (design -> evaluation) and direct optimization (BFGS) baseline. The authors claim, and the experimental supports, that the proposed two-stage method has a performance advantage of direct learning and a compute advantage, in inference, over direct optimization. The authors also discuss how even the training cost for their method is higher compared to direct optimization, the training cost gets amortized in inference as the two-stage does not require re-training/re-running like direct optimization.