Deep Reasoning Networks:  Thinking Fast and Slow, for Pattern De-mixing
Authors: Di Chen
Yiwei Bai
Wenting Zhao
Sebastian Ament
John M. Gregoire
Carla P. Gomes
Venue:  (2020)
Keywords: Deep Reasoning Network
Pattern De-mixing

Abstract: We introduce Deep Reasoning Networks (DRNets), an end-to-end framework that combines deep learning with reasoning for solving pattern de-mixing problems, typically in an unsupervised or weakly-supervised setting.  DRNets exploit problem structure and prior knowledge by tightly combining logic and constraint reasoning with stochastic-gradient-based neural network optimization.  We illustrate the power of DRNets on de-mixing overlapping hand-written Sudokus (Multi-MNIST-Sudoku) and on a substantially more complex task in scientific discovery that concerns inferring crystal structures of materials from X-ray diffraction data (Crystal-Structure-Phase-Mapping). DRNets significantly outperform the state of the art and experts' capabilities on Crystal-Structure-Phase-Mapping, recovering more precise and physically meaningful crystal structures. On Multi-MNIST-Sudoku, DRNets perfectly recovered the mixed Sudokus' digits, with 100% digit accuracy, outperforming the supervised state-of-the-art MNIST de-mixing models.

Under review as a conference paper at ICLR 2020
DEEP REASONING NETWORKS FOR UNSUPERVISED PATTERN DE-MIXING
Anonymous authors Paper under double-blind review
ABSTRACT
We introduce Deep Reasoning Networks (DRNets), an end-to-end framework that combines deep learning with reasoning for solving pattem de-mixing problems, typically in an unsupervised or weakly-supervised setting. DRNets exploit problem structure and prior knowledge by tightly combining logic and constraint reason- ing with stochastic- crraatti-ggrdiiet-bbaeee neural network optimization. We illustrate the power of DRNets on de-mixing overlapping hand-written Sudokus (Multi- MNIST-Sudoku) and substantially more complex task in scientific discovery that concerns inferring crystal structures materials from X-ray diffraction data Mapping). DRNets significantly outperform the state of the art and experts' capabilities on Eryystl-Structtrr Caciuure-Phaeeaaae Phase-Mapping, recovering more precise and physically meaningful crystal structures. On Multi-MNIST- Sudoku, DRNets perfectly recovered the mixed Sudokus' digits, with 100% digit accuracy, outperforming the supervised state-of-the-an MNIST de-mixing models.
(Crystal-Struectt -itr--Stttuue-euuie Phaaeraaaaaa---iiggg....
INTRODUCTION
Deep learning has achieved tremendous success in areas such vision, speech recognition, language translation, and autonomous driving. Nevertheless, certain limitations of deep learning are generally recognized, in particular, limitations due the fact that deep learning approaches heavily depend on the availability of large amounts of labeled data. In certain domains, such as scientific discovery, it is often the case that scientists don't have large amounts of labeled data and instead have rely on prior knowledge to make sense of the data. One grand challenge in scientific discovery is to perform thghthhroughput unsupervised interpretation of scientific data, given its exponential growth in generation rates, dramatically outpacing humans' ability to analyze them. Herein we consider pattern de-mixing problems, which involve decomposing a mixed signal into the collection of source patterns, such separating mixtures of X-ray diffraction (XRD) signals into the source XRD signals of the corresponding crystal structures, a key challenge in materials discovery. More generally, pattern de-mixing problems are pervasive in scientific areas diverse as biology, astronomy, and materials science, as well as in commercial applications for e.g., healthcare and music.
We propose Deep Reasoning Networks (DRNets), an end-to-end framework that combines deep learning with logical and constraint reasoning for solving unsupervised or very-weakly-supee supervvsed pattern de-mixing tasks. We illustrate the power of DRNets for disentangling two overlapping hand- written Sudokus (Multi-MNIST--udookk de-mixing task in scientific discovery that concerns inferring crystal structures of materials from X-ray diffraction data, which we refer to as require probabilistic reasoning interpret noisy and uncertain data, while satisfying a rules: Sudoku rules and thermodynamic rules, respectively. For example, de-mixing hand written digits is challenging, but becomes more feasible when we reason about the prior knowledge concerning the two overlapping Sudokus. Crystal structure phase mapping yet substantially more complex. In fact, crystal structure phase mapping easily becomes too complex for experts to solve and is major bottleneck in high-throughput materials discovery. DRNets inspired and motivated by problems from scientific discovery, such as crystal structure phase mapping.
(see Fig and for solving substantially more complex
Strructure-Paae-
Both de- de-mixing tasks
Our contributions: (1) We introduce Deep Reasoning Networks (DRNets), an end-to-end frame- work that combines deep learning with logical and constraint reasoning for unsupervised or very- weakly-supervised de- de-mixing tasks. Specifically, DRNets perform end-to-end deep reasoning by

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The two ground-truth Sudokus
(b) Imput mixture
DRNet de-mixed Sudokus (d) Reconstructed mixture
Figure (a) Two 4x4 Sudokus: The cells in each row, column, and any of the four 2x2 boxes
involving the corner cells have non-repeating digits. (b) Two overlapping Sudokus, with mixture of two digits in each cell: one from to 4 and the other from to In Multi-MNNIST-Sudokk the digits of two overlapping hand written Sudokus (b) have to be de-mixed (as done by DRNets in (c)). (d) The reconstructed overlapping hand written Sudokus from DRNets.
Prototypes pes
Input Mixed Digits
De-mixed Digits
Reconstructed Mixtures
Structured Latent Space
Reconstruction
Loss
Generative Decoder (cond-Gan)
BBOT BOOG
Overall Objective
Encoder
Reasoning
Loss
Overlapping Sudokus
Local Constraints
Global Constraints
Batching Controller
Weights Controller
(c)
Kuonledge
Reasoning Modules
(a)
(b)
Figure 2: Deep Reasoning Networks (DRNets) perform end-to-end deep reasoning by encoding a latent space of the input data that captures prior knowledge constraints and is used by generative decoder to generate the targeted output. (a) Prior knowledge includes prototypes of digits, which used to pre-train and build the decoder's generative module, and Sudoku's 's rules, which help DRNet reason about the overlapping digits. (b) Reasoning modules batch data points involved in the same constraints (cells in rows, columns, blocks of Sudoku) together, enforce that the structure of the latent space satisfies prior knowledge, and dynamically adjust the weights of constraints based (c) The overall objective combines responses from the generative decoder (thinking fast) and the reasoning modules (thinking slow).
on their satisfiability.
encoding a latent space of the input data that captures the structure and prior knowledge constraints within and among data points (F The latent space used by a generative decoder to generate the targeted output, which should be consistent with the input data and prior knowledge. Subse- quently, DRNets optimize an objective function capturing the overall problem objective as well as prior knowledge in the form of weighted constraints. (2) To instantiate the logical constraints in DRNets, we introduce a group of entropy-based continuous relaxations that use probabilistic modeling to encode general discrete constraints including sparsity, cardinality and so-called All- Different constraints. To optimize those constraints, we introduce a variant of standard SGD method (Robbins & Monro 1985) called constraint-awarr stochastic gradient descent, which batches data points involved in the same constraint component together and dynamically adjust the constraints' weights as function of their satisfiability. In the following sections, we show how encode WEI---NNIST-SSuook and Erystal-Structuun CeetttrrePhaae--aaee Chase-Mapping as DRNets, by properly defining the structure of the latent space, additional reasoning modules model the problem constraints (prior knowledge), and the components of the objective function. De facto, these examples illustrate how to develop "gadgets" to encode a variety of constraints and prior knowledge in DRNets. (3) We demonstrate the potential of DRNets on two de-mixing tasks with detailed experimental results. We show how (3.1) DRNets significantly outperformed the state of the art and human experts on SesttalStructure-haaae Phase-Mapping instances, recovering more precise, interpretable, and physi- cally meaningful crystal structure pattern decompositions. In this task, DRNets solve a previously unsolved chemical system, which subsequently led to the discovery of material that is important
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for solar fuels technology. (3.2) On Multi-MNIST-Sudokt instances, without direct supervision, DRNets perfectly recovered the digits in the mixed Sudokus with 100% digit accuracy, outperforming the supervised oftte-of-thhearr MNIST de-mixing models, including CapsuleNet (Sabour et al. 2017) and ResNet (He et al. 2016).
2 RELATED WORK
DRNets have been motivated by scientific tasks such as crystal phase mapping that involve identifying or de-mixing patterns in data that satisfy prior scientific knowledge. In general, for such tasks there are no labeled datasets. So our work focus on unsupervised or weakly supervised learning, using prior knowledge.
Most closely related work: Unsupervised or weakly supervised de-mixing approaches. Pattern de-mixing approaches have been developed under the name of source separation in the signal processing community. The unsupervised methods in this area mostly try to solve the de-mixing, which is in general ill-posed, using different regularizations. Among existing methods, recent work for weakly supervised audio source separation (Zhang al. 2017) most related to DRNets since they also employed a generative adversarial network (GAN) in their model. However, their model mainly employs the decoder of GAN to discriminate the reality of separated sources, while DRNets only utilize the generator of GAN as the generative model of possible sources. Moreover, the weakly supervised setting in their paper is actually too strong: they need the true labels of mixed sources, which is almost the goal of our tasks, and therefore it is not applicable to our settings. We now consider the sfate-of-the ofr models for the tasks considered in this paper. For Crystal-structure- thase-mapping due to the lack of labeled datasets, existing models (Ermon et al.] 2015) Xue jet al. 2017 [aai et al.) 2017 2018 Stanev et al., 2018) are mainly based on non-negative matrix factorization (NMF), which is in general unsupervised. (Stanev et al. (2018) proposed the NMF-k algorithm, which applics a customized clustering process over the results of thousands of runs of pure NMF algorithm (Long et al. 2009) to cluster the common phase patterns. However, NMF-K does not enforce prior knowledge (namely thermodynamic rules) and therefore the solutions produced often not completly physically meaningful. To address this limitation several approaches have been developed that use external de-mixing module to enforce prior knowledge (Ermon al. 2015 Bai et al. 2017 2018). However, the coordination barrier between the NMF de-mixing module and the reasoning module often results in inferiror performance, where the solution satisfies constraints at the cost of huge reconstruction loss. In contrast to existing models, DRNets seamlessly integrate the pattern de-mixing modulc and the reasoning module, recovering almost exact ground truth decomposition. In our experiments we thoroughly compare DRNets performance against the state of the art (IAFD and NMF-k) for crystal-structues
integer programming modules to interact with the NMF
pattern de-mixing. MNIST de-mixing was first studied by |Hinton et al.) in 2000 where the aim is to identify or de-mix overlapping digits coming from the MNIST datasets (LeCun et al. 1998). More recently, has been tackled with state-of-the- as CapsuleNet (Sabour et al. 2017) and ResNet (He et al., 2016). Existing works concerning this task are mainly in supervised settings, where we have labels of digits for each overlapping image. However, in this paper, aim tackle this task in weakly supervised setting, where we only have access to the prototypes of single digits and the extra Sudoku rules. Due to the lack of existing models with the same setting, we compared DRNets's performance against the state-of-the-an supervised models (CapsuleNet and ResNet). By utilizing the supervision from prior knowledge and reasoning, we show that DRNets' outperformed all supervised models with 100% digit accuracy.
neural network models such
Enhancing deep learning with symbolic prior knowledge. Exploiting problem structure and reasoning about prior knowledge has been of increasing interest facilitate deep learning (Garcez ct al.) 2019). In computer vision, symmetry constraints, bonc-length constraints and linear constraints were introduced for human pose estimation (Zhou et al. 2017 2016) and image segmentation (Pathak et al. 2015) to regularize the output and enhance generalization. In natural language processing, Hu ctt al. 22166a b) introduced the posterior regularization (Ganchev et al., 2010) framework into deep learning to incorporate rule-based grammatical knowledge using first order logic. Xu et al. (2017) proposed a semantic loss function to enforce propositional logic constraints on the output of neural networks for semi-supervised multi-class classification tasks. Wang et al. (2019) proposed SATNet, which approximately encodes a MAXSAT solver into a neural network layer called SATNet layer, to explicitly learn the logical structures (e.g.. parity function and Sudoku) from the labeled training data.
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Previous works in this area primarily focus on supervised semi semi-supervised settings for data-rich domains, where direct supervision from labels reduce the importance of explicitly reasoning about prior knowledge. In contrast, with an unsupervised setting, the supervision of DRNets comes from reasoning about prior knowledge and seff-reconstrucctoon, which is strongly desired for problems in scientific discovery due to the lack of labeled datasets, and strongly motivated by extensive prior knowledge from sources ranging from fundamental principles to the intuitive experience of scientists.
Among existing works, SATNet is mostly related to DRNets in the sense of bridging logical reasoning with deep learning. However, SATNet is essentially designed for learning logical structures (prior knowledge) from labeled training examples while DRNets aim to facilitate unsupervised learning with known logical constraints. In terms of the encoding of the reasoning module, the semantic loss (Xu al. 2017) is mostly related to ours. However, the semantic loss encodes constraints by propositional logic, which requires enumerating all possible Boolean assignments that satisfy the constraints. Consequently, the semantic loss has to enumerate a large number of assignments to encode constraints such as k-sparsity constraints and All-Different constraints, which is not applicable to tasks considered in this paper.
DEEP REASONING NETWORKS
Oplimization with discrete and continuous variables
Data-Driven Constrained
Lagrangiox Relaxatio
Data-Driven Unconotrainel Optimization with discrete and continuons variables
Continnovy Relavation
Data-Driven Unconstrained Optimiiation with continuous vuriables
Constriiut-bwree SGD
Deep Reasoming Networks
Figure 3:
The reduction flow of Deep Reasoning Networks.
DRNets (see Fig 2 are inspired by human thinking (Shivhare & Kumar 2016): we abstract patterns to higher-level descriptions and combine them with prior-knowledge to fill-in the gaps. Consider the Muli-MNNIST-Sudoki example (Fig. I): we first guess the digits in each cell based on the patterns; we re-adjust our initial beliefs and re-image the overlapping patterns by reasoning about Sudoku rules and comparing them to the original ones, potentially involving several iterations. Analogously, in reasoning system, inference procedure derives what follows from initial set of axioms and rules. For example, in standard 9x9 Sudoku, an inference procedure identifies the missing cell values of the input Sudoku. A constraint solver is a particular type of reasoning system in which axioms and rules are expressed as constraints and the inference procedure is a search method.
Formally, DRNets formulate unsupervised pattern de-mixing as a data-driven constrained op- timization, incorporating abstractions and reasoning about structure and prior knowledge:
Σ C(C(do(x:)), x:) S.L. (x::) E glocall and (6))))))....
(x(x)) E g2global
min
(1)
N
In this formulation, x; E R" is the i-th n-dimensional input data point, (-) is the function of the encoder in DRNets parameterized by 8, G(-) denotes the generative decoder, -) is the loss function (c.g.. evaluating the reconstruction of patterns). 52 Ilocal and 52global are the constrained spaces w.r.t. a single input data point and several input data points, respectively. G(-) general a fixed pre-trained or parametric model. For example, in Multi-MNIST-Sudokku G(.) is a pre-trained conditional GAN
G(-) a Gaussian Mixture model. Note that constraints can involve several (potentially all) data points: e.g., in Sudoku, all digits should form a valid Sudoku and in crystal persstalstructurephiie watee-hhaaapaeeeeeeee mapping, all data points in composition graph should form valid phase diagram. Thus, we specify local and global constraints in DRNets - local constraints only involve a single input data point whereas global constraints involve several input data points, and they are optimized using different strategies.
(Mizaa& & Osindero 2014) using hand-written digits, and for Crystal- Seiecture-Phaee-Meee
Solving the constrained optimization problem IT) directly extremely challenging since the objective function in general involves deep neural networks, which are highly non- non-linear and non-convex, and prior knowledge often even involves combinatorial constraints (Fig Therefore, we use Lagrangian relaxation to approximate equation (1) with an unconstrained optimization problem, i.e.,
N N Σ C(C((v(x))))x) N' (do(x:)) Σ (L S2))
mim
N is the number of input data points, N, denotes the number of global constraints, Sy denotes the set of indices w.r.t. the data points involved in the j-th global constraint, and vfl v/f denote the penalty
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functions for local constraints and global constraints, respectively, along with their corresponding penalty weights X' and X/. In the following, we propose two mechanisms to tackle the above unconstrained optimization task (Fig
Continuous Relaxation: Prior knowledge often involves combinatorial constraints with discrete variables that are difficult to optimize in an end-to-end manner using gradient-based methods. Therefore, we need to design proper continuous relaxations for discrete constraints to make the overall objective function differentiable. Existing works (Hu et al. 201 Xu et al. (017)) proposed several relaxations for injecting first-order logic and propositional logic into deep learning. However, limited by the expressive power of those logic formulas, we need a large number of logical terms to express constraints such as k-sparsity constraints or All-Different constraints. Therefore, to instantiate DRNets for our tasks, we propose a group of untropy-based continuous relaxations to encode general discrete constraints such sparsity, cardinality and All-Different constraints (see We construct continuous relaxations based on probabilistic modelling of discrete variables,
Fig 4).
Cardinality Constraint
Cardinality Constraint Relaxation 0
EL {0,1) j= 1...8 s. Σ}=1 ft.) = and E 11
HP; + H(Q;)
= log Pt, - - Qi, log Quj
All-Different Constraint
All-Different Constraint Relaxation
For all constrained set
For all constrained set S & MaxH(Ps) + H(Qs)
s.t. Eies Eij = for ...
k- k-Sparsity Constraint
k-Sparsity Constraint Relaxation mjn (P))) where c < log
eLj [0,1] j= 1 s. S. e k
Figure 4: Examples of continuous relaxations: eij P., Qs Pas denote binary variables, the discrete distribution over digits I to 4, the discrete distribution over digits 5 to 8, and the discrete distribution over values to M.
where we model a probability distribution over all possible values for each discrete variable. For example, in Multi-MNIST-SSuokuu a way of encoding the possible two digits in the cell indicated by data point v; (one from (1..4) and the other from (5...8)). ). is to use 8 binary variables ei2j E (0, 1}, while requiring Σ}= fx,) and E = In DRNets, we model probability distribution P; and Qi over digits 1 and to 8 respectively: P..j.j-1...4 and ii..j= denote the probability of digit j and the probability of digit 4, respectively. We approximate the cardinality constraint of ei by minimizing the entropy of P; and Q, which encourages P: and Q; to collapse to one value. is the All-Different constraint, where all
Another combinatorial constraint in Mult--MNIST-Suude the cells in constrained set S. i..., each row, column, and any of four 2x2 boxes involving the corner cells, must be filled with non-repeating digits. For probabilistic relaxation of the All-Different constraint, we analogously define the entropy of the averaged digit distribution for cells constrained set S, i.e., (Ps)):
H(Ps) = Ps. log Ps.s = Σ
Ps.
Σ iES
Po.
log
Σ Pes
(3)
S
In this equation, larger value implies that the digits in the cells of S distribute more uniformly. Thus, we can analogously approximate All-Different constraints by maximizing H (Ps)) and H (Qs). One can see, by minimizing all H(P.) and (Q.) to as well as maximizing all ((Ps) and H (Qs) to log [S], we find a valid solution for the two 4x4 Sudoku puzzles, where all P.,j are either 0 or 1.
We also relax k- k-spariity constraints, which for example in Crystal- Phase- Maappig state the maximum number l: of pure phases in an XRD-pattem, by minimizing the entropy of the phase distribution PM below a threshold < log We choose the threshold e < log because the entropy of discrete distribution PM concentrated on at most & values cannot exceed log k. Note that other relaxations can be adapted in DRNets, for these and other tasks. See also additional relaxations (c.g., for SAT constraints). detailed relaxation derivations, and implementation details in supplementary materials.
We introduce a variant of standard SGD method called constraint-ad aware SGD, which is conceptually similar to the optimization process in GraphRNN (You et al. 2018). to tackle the optimization of global penalty functions w ((do (xx)]k E Sy}), which involve several (potentially all) data points. We define constraint graph, an undirected
Constraint-Awan Stochastic Gradient Descent:
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Algorithm 1 Constraint-aware stochastic gradient descent optimization of deep reasoning networks.
Input: (i) Data points {ar (ii) Constraint graph. (iii) Penalty functions (.) and v))(() for the
local and the global constraints, (iv) Pre-trained or parametric generative decoder G(.).
1: Initialize the penalty weights X', X2 and thresholds for all constraints.
2: for number of optimization iterations do
3: Batch data points {x1,...; Xm from the sampled (maximal) connected components.
4: Collect the global penalty functions (-) concerning those data points.
5: Compute the latent space (xo(x)),
(Xm)) from the encoder. 6: Adjust the penalty weights λτ, X3 and thresholds accordingly.
7: minimize 1
in - ((G((o(x;) ;,,)) x;) + Ayos (()g()))) + شيا (foo((() e s,))
using any standard gradient based optimization method and update the parameters A.
8: end for
graph in which each data point forms a vertex and two data points are linked if they are in the same global constraint. Constraint- SGD batches data points from the randomly sampled (maximal) connected components in the constraint graph, and optimizes the objective function w.r.t. the subset of global constraints concerning those data points and the associated local constraints. For example, in Multi-MNiSST-udonn we batch the data points from several randomly sampled overlapping Sudokus and optimize the All-Different constraints (global) as well as the cardinality constraints (local) within them. However, in Crystal Structure- PhaseePPaaaaaee Mapping, the maximal connected component becomes too large batch together, due to the constraints (phase field connectivity and Gibbs- alloyyng rule) concerning all data points in the composition graph. Thus, we instead only batch subset (still a connected component) of the maximal connected component e.g., a path in the composition graph, and optimize the objective function that only concerns constraints within the subset (along the path). By iteratively solving sampled local structures of the "large" maximal component, we cost-efficiently approximate SGD
each overlapping Sudoku forms a maximal connected component,
Crystal-Structure-
the entire global constraint. Moreover, for optimizing the overall objective, constraint- dynamically adjusts the thresholds and the weights of constraints according to their satisfiability. which can involve din-differenti functions (See details in appendix). For efficiency and potential capability of generalization, DRNets solve all instances together using constraint-aware SGD (see Algorithm
4 EXPERIMENTS
We illustrate the power of DRNets mainly on two pattern de-mixing tasks disentangling two over- lapping hand-written Sudokus (Nulti-NNIST-Sudoku) and inferring crystal structures of materials from X-ray diffraction data details of the experiments and the experimental results of DRNets on other tasks in supplementary
Limited by the space, we put the material. Note that, since DRNets an unsupervised framework,
apply the restart (Gomes) et al. 1998) mechanism, i.e., we can re-run DRNets for unsolved instances.
P1a
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Method
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98.0
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CapsoleNer
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((%,.)
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$7.8
Rimins
ResNot-18
97.7
68.5
Jmint JOhrs
ResNet-18 + local search
97.7
$8.3
3bese 10hrs
Figure 5: Left: The latent space of DRNets for Piilii-MNIST-Sudeeee Right: Accuracy comparison. show "test time + training time" for supervised baselines and the generalization mode of DRNet, and "solving time" for the optimization mode of DeNets. (See also supplementary materials.)

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Multi-MNIST-Sudoou: We generated 160,000 input data points for each training set, validation set and test set, where each data point corresponds to a 32x32 image of overlapping digits coming from MNIST (LeCun et al., 1998) and every 16 data points form 4-by-4 overlapping Sudokus. For Multi-MNIST SuuuuuNi DRNets batch every 16 data points together to enforce the All-Different constraints among the cells of each Sudoku. The encoder of DRNets is composed of two ResNet- 18 He et al. (2016) and we use a conditional GAN (Mirza & Osindero 2014) as our generative decoder (denoted as G())), which is trained using the digits in the training set of MNIST. For each cell X;, the encoder encodes a latent space, which consists of two parts: The first part includes two distribution P, and Qi (see F concerning the possible digits in the cell, and the second part is the latent encodings =2,1 ..... Fi.8 of each possible digit conditioned on the overlapping digits, which used by the generative decoder to generate the corresponding digits G(z,.)). We estimate the two digits in the cell by computing the expected digits over P. and Qi i.e., - G(z.( and
e G(z.G4z))))) and reconstruct the original input mixture (see Fig 5). As described above, we impose the continuous relaxation of the cardinality and All-Different constraints to reason about the the Sudoku structure among cells of the overlapping Sudokus. To demonstrate the power of reasoning. we compared our unsupervised DRNets with supervised start-of-the-aa art MNIST de- m-xiiix models CapsuleNet (Sabour et al., 2017) and ResNet (He et al.) (2016), and variant of DRNets that removes the reasoning modules ("DRNets w/o Reasoning") ". To saturate the performance of baseline models, also applied a post-process local search for them to incorporate the Sudoku Rules. Specifically, we did a local search for the top-2 (top-3 would take too long to search) most likely choice of digits for each Sudoku of the two overlapping Sudokus and try to satisfy Sudoku rules with minimal modification compared with the original prediction. We evaluate both the percentage of digits that are correctly de-mixed (digit accuracy) and the percentage of overlapping Sudokus that have all digits correctly de-mixed (Sudoku accuracy). Empowered by reasoning. DRNets significantly outperformed CapsuleNet, ResNet, and DRNets without reasoning, perfectly recovered all digits with the restart mechanism (see Fig 5) and additionally reconstructed the mixture with high-quality (see Fig [. Moreover, because DRNets solve all instances together (see Algorithm 2 not only can DRNets solve instances directly on the test set from random initialization, DRNets can also generalize from the training set to test set, given enough training examples. DRNets learn to generalize its the test set by solving the training set instances self-supervised (Jing & Tian| 2019) by Sudoku rules, instead of labels, and even outperform CapsuleNet and ResNet (Fig 5) Note that, for unseen instances in the test set, we further optimize the instances for 25 steps to achieve the reported performance (Additional details in the supplementary material).
de-mixing performance
Input:: XRD patterns
Encoder Pai K
0.93
G(Zu)
in composition graph
E()
Piz
2,2
0.06
G(2z)
XRD Mcasurement X
Thenodggaanis
Gibt Rak
PiM
0.01
G(Zu)) DRNef Reconstructed XRD
M P
G(Zu)
Allesing
sampled
Phave Ceeertite
Generative Decoder
Generated Phuse Patterns
Measurement
path
ICDD slick pattems
G() (GMM)
Figure 6: The latent space of DRNets for Crystal- PenettrrePPaae-MMeee Chapennapping.. M denotes the number of possible phases. (For AI-Li-Fe, M = 159; For Bi-Cu-V, M = 100.)
cystal-Strucctur-PPae Phase-Mappingg concerns inferring crystal structures from set of X-ray diffrac- tion measurements (XRDs) of a given chemical system, satisfying thermodynamic constraints. Crystal structure phase mapping is very challenging task, major bottleneck in high-throughput materials discovery: Each X-ray measurement may involve several mixed crystal structures; each chemical system includes hundreds of possible crystal structures; for each crystal structure pattern, we only have a theoretical (idcalized) model of pure crystal phases; the thermodynamic rules are also complex; and the crystal patterns are difficult for human experts to interpret. Herein, we illustrate DRNett for crystal structure phase mapping for two chemical systems: (1) ternary AI-Li-Fe oxide system (Le Bras et al. 2014)), which is theoretically based, synthetically generated, with ground truth solu- tions, and (2) ternary Bi-Cu-V oxide system, which is more challenging real pneriment-based system, more noisy and uncertain. For each system, each input data point is the XRD of a mixture of crystal structures. Additionally, the input includes the composition graph specifying elemental compositions and the constraint graph of the data points. We also collected a library of possible crystal structures from the International Centre for Diffraction Data (ICDD) database. Fach crystal
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structure (also named phase) is given list of diffraction peak location amplitude pairs, (referred to as stick pattern), representing the ideal phase patterns measured in a perfect condition (see Fig 6) To model more realistic conditions, DRNets simulate the real phase patterss from stick patterns using Gaussian mixture models, where the relative peak locations and mixture coefficients are given by the stick locations and amplitudes. Moreover, the peak width, peak location shift, and peak amplitude variance are parameterized by the latent encoding 2i,j and used by the generative decoder to generate the corresponding possible phase patterns in the reconstructed XRD measurement.
Phase Phose 2 Phase 3 Phose Phose Phase
Chemical Systems:
Recomsrration Losses
Phuse Fidelity Loss
Thermodynamic Kules Satisfuction
NMF-k IAFD DRNets Groend 1rath
(Percentage data pphse Reld
conutriin))
AILI-Fe
Li Loss
L22
JS distince (x30 3)
Gibbs
Gibbs Allay
Counecivityy
Loss
DKNNets
0,839
<0.001
-11801
100%
100%
100%
JAFI
3,990
0.535
11.30
100%
100%
100%
NMFA
56.10
Bi-Cu-V
L1 Loss
Loss
1.2
15 distance (x30 +)
Gibbs
Gibbs
Allay
Comneciviiy
Plak
DRNess
3,993
0.196
8,370
100%
100%
100%
IAFD
7.425
0.545
93.36
100%
99
95%
NMF k
8.033
0.655
92.63
51%
35%
83%
Figure 7: Left: Comparison of phase concentration and reconstruction loss for different methods in AI-Li-Fe oxide system. Note that, pure phases (out of 159 possible candidates) appear in the system and result in 15 different mixtures. Each dot represents an XRD measurement whose size proportional to the estimated phase concentration. DRNet's phase concentration closely match the ground truth in contrast to IAFD's and NMF-k's. The heatmap on the right shows that DRNets reconstruct the XRD measurements much better than other methods with respect to the L.1 loss. Right: DRNets outperform both IAFD and NMF-k with better reconstruction error and perfect rule satisfaction on both systems. (additional details for Bi-Cu-V in the supplementary material).
We compared DRNets with IAFD Baii et al. 2017) and NMF-k (Stanev et al., 2018), which are both state-of-the-ae aht non- non-negative matrix factorization (NMF) based unsupervised de-mixing models. NMF-K improves the pure NMF algorithm (Long et al. 2009) by clustering common phase patterns from thousands of runs. However, NMF-K does not directly enforce thermodynamic rules and therefore the solutions produced are often not completely physically meaningful. IAFD uses external mixed-integer programming modules to enforce thermodynamic rules during the de-mixing. However, due to the gap between the external optimizer and NMF module, the solution of IAFD is still far from the ground truth. Our evaluation criteria include reconstruction losses, phase fidelity loss and the satisfaction of thermodynamic rules. Note that, the phase fidelity loss measures the JS distance between the de-mixed phases and the closest ideal phases by fitting the de-mixed phases with the ICDD stick patterns using the physical model (Le Bras et al. 2014). As shown in Fig 7 for the AI-Li-Fe oxide system, the phase concentration (the distribution of de-mixed pure phases over all data points of that chemical system) of either IAFD NMF- is far from the ground truth. In contrast, DRNet almost exactly recovered the ground truth solution by seamlessly integrating pattern recognition, reasoning and prior knowledge. Moreover, by explicitly incorporating the ICDD stick pattern information into DRNets, the phases de-mixed by DRNets are much more real than those from IAFD and NMF- (see phase fidelity loss). For Bi-Cu-V oxide system, DRNets solved this previously wnsolved real system, producing valid crystal structures and significantly outperforming IAFD and NMF- w.r.t. reconstruction errors and phase fidelity loss. In addition, materials science experts thoroughly checked DRNet's solution of Bi-Cu-V oxide system, approved it, and subsequently discovered a new material that is important for solar fuels technology.
5 CONCLUSIONS AND FUTURE WORK
We propose DRNets, a powerful end-to-end framework that combines deep learning with logical and constraint reasoning for solving unsupervised pattern de-mixing tasks. DRNets outperform the state of the art for de-mixing MNIST Sudokus and crystal-structur phase mapping, solving previously unsolved chemical systems substantially beyond the reach of other methods and materials science experts' capabilities. While we illustrate the potential of DRNets with unsupervised settings, it is straightforward to impose supervision into DRNets. Future research includes exploring DR
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for incorporating other types of constraints, prior knowledge, and objective functions, for other applications.
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A SUPPLEMENTARY MATERIALS
Herein, provide additional details about DRNets and experimental settings for better understanding of DRNets and reproducibility of our results. Code and datasets to reproduce the experiments will be provided with the final version of the paper.
A.1 CONTINUOUS RELAXATION
In this section, we provide more relxations for other constraints such as SAT constraints and provide an intuitive high-level informal proof that all the relaxations converge valid solution of the discrete version when it achieves its minimal value. Fig g summarizes the relaxations.
Cardinality Constraint
Cardinality Constraint Relaxation
e1, E [0.13 == 1...8 t. Σj=1 St, = and -1y
mjn H(P)) + H(Q:)
0 - log P., - Σ Qu lig
All-Different Constraint
All-Different Constraint Relaxation
For all constrained set S s. Eiese,, f1) 1 for 1 ...
For all constrained set S max (P + 11(ss)
k-Sparsity Constraint
k-Sparsity Constraint Relaxation mjn max(((Pw).c)) where logk
eij (0.1) j= 1 M..M s. ,t.E)=-e,, Integer Programming Encoding of SAT For any literal x2 and its negation & s. s.t.x, {0.1) and x; + X; For any clause Cj = dz V... djx,
SAT Relaxation
any literal x; and negation X; = 1 1...N). ) we model
distribution B, Bern 9t), s.t. x; _ pi. and x; a qi
For all clause Cj = a,,1 V O,K,,- j ...Ncc min E/= leaky_relu (1 1 91x) + Ao E H(B)
Vayx & s. I aqxx 21
Figure 8: Examples of continuous relaxations: Et,j Pe, Qú, Pas Ne. Ni, Kj, An B, denote binary variables, the discrete distribution over digits 1 to 4, the discrete distribution over digits 5 to 8, the discrete distribution over values M, the number of clauses, the number of literals, the number of literals in the j-th clause, the weights of entropy terms, and the Bernoulli distribution for the i-th literal. "leaky relu" the leaky ReLU,
For cardinality constraints, when the entropy of distribution P; and Q: reaches 0, all the probability mass collapses to only one variable. Therefore, all Pt, and Qi, are either which valid solution of the original discrete constraints.
For All-Different constraints, maximize the entropy of the averaged digit distribution for all cells in constrained set S, i.e., H (Ps). Note that, the All-Different constraints are imposed together with the cardinality constraints. Therefore, when the entropy of the digit distribution in each cell zero, we know that the digit distribution of each cell converges to one digit. Hence, if H (Ps) reaches its maximum, i.e., log (),, we have [S] Eies Ptj [ST for all digit j. Crossed with the fact that Pr,; are either 0 or when the cardinality constraints are satisfied, we know that only one Pt, is equal to 1 for all cell in the set S and others are 0, which directly states the All-Different constraints.
We derive the k- Sparsity constraints in a similar way as the cardinality constraints except that we want to force the distribution to concentrate at most k digits. By normalizing the values of

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discrete variables ei,j (j I...MM to a discrete distribution PM, we minimize the entropy of distribution PM to at most log k. which is the maximal entropy when the distribution concentrates on only & values. Though, H HPM)) log k is not sufficient condition for k-sparsity, we can initialize the threshold e of k-sparsity constraints to log k and dynamically adjust the value of e based on the satisfaction of the k-sparsity constraints. In practice, it works well with the supervision from other modules, such as the self-reconstruutiion
For SAT constraint relaxations, the key idea is to minimize the entropy of the Bernoulli distribution over each literal to force it converge to either 1 or O. Then, we maximize the sum of the value of literals in each clause (or their negation) to encourage one of the literals be However, maximizing the sum of the value of literals does not necessarily give you valid assignment because there could exist an assignment that the sum of literals in some clauses are O and the sum of literals in other clauses are very large. Therefore, we use leaky_rule (Xu et al., 2015) function to discount the loss when the sum is larger than 1. As shown in F we formulate the relaxation loss function in a form to be minimized. For K-SAT problems with No clauses, we can set the leaky ratio to be N. so that any invalid assignment cannot have loss that is less or equal to O. On the other hand, for any valid assignment, the sum of literals in each clause is at least 1. Thus, we can obtain a valid assignment of k-SAT constraints by minimizing the loss function to O.
We describe other task specific constraints (c.g., phase field connectivity constraints) in the following experimental sections.
A.2 CONSTRAINT-AWAR STOCHASTIC GRADIENT DESCENT:
Algorithm 2 Constraint-aware stochastic gradient descent optimization of deep reasoning networks.
Input: (i) Data points ] (ii) Constraint graph. (tii) Penalty functions (-(-) and why( (-) for the
local and the global constraints. (iv) Pre-trained or parametric generative decoder G(.).
1: Initialize the penalty weights X', Xÿ and thresholds for all constraints.
2: for number of optimization iterations do
3: Batch data points {x,, } } from the sampled (maximal) connected components.
4: Collect the global penalty functions (-)}²² concerning those data points.
5: Compute the latent space (xo(x))...) (x)).... dg (xm)) from the encoder.
6: Adjust the penalty weights Åτ, X and thresholds accordingly.
7: minimize 1 (E L(G(ába (x;)): x;) + (cbo(()) + Σラ (100(()) e S))))
using any standard gradient- based optimization method and update the parameters O.
8: end for
We introduce a variant of standard SGD method called constraint-awarr SGD, which is conceptually similar to the optimization process in GraphRNN (You et al., 2018), to tackle the optimization of S, which involve several (potentially all) data points. We define a constraint graph, an undirected graph in which each data point forms a vertex and two data points are linked if they are in the same global constraint. Constraint-aware SGD batches data points from the randomly sampled (maximal) connected components in the constraint graph, and optimizes the objective function w.r.t. the subset of global constraints concerning those data points and the associated local constraints. For example, in Multi-MNISTTSudokk. each overlapping Sudoku forms a maximal connected component, we batch the data points from several randomly sampled overlapping Sudokus and optimize the All-Different constraints (global) as well as the cardinality constraints (local) within them. However, in -trusal-Struturee.. maximal connected component becomes too large to batch together, due to the constraints (phase connectivity and Gibbs-alloying rule) concerning all data points in the composition graph. Thus, we instead only batch subset (still connected component) of the maximal connected component - c.g.. a path in the composition graph, and optimize the objective function that only concerns constraints within the subset (along the path). By iteratively solving sampled local structures of the "large" maximal component, we cost-efficiently approximate the entire global constraint.
global penalty functions vi ({ (foo( (xx))]k (x
Mapping. the
Moreover, for optimizing the overall objective, constraint-aware SGD dynamically adjusts the thresholds and the weights of constraints according to their satisfiability, which can involve non- differentiable functions. Specifically, we initialize penalty weights of constraints and thresholds for
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penalty functions using paper-parmettrss. During training, we check the satisfiability of constraints (this step could involve non non-differentiable functions) after several epochs and increase the penalty for violated constraints. For example. the threshold of k-sparsity is initialized as log k, which is the entropy of the case that the probability mass is evenly distributed among k entities. Thus, it could be the case that there are more than i entities, but their probability mass is not evenly distributed. Hence, we check the satisfiability of k-sparsity constraint: if the entropy is already below the current threshold (log k) and there are still more than k entities with probability mass more than e (0.01), we decrease the threshold to keep enforcing the model to minimize the entropy to reach the k-sparsity,
Finally, to better exploit parallelization, DRNets solve all instances together using constraint-awar SGD (see Algorithm
A.3 RESTART MECIIANISM FOR DRÑETS:
Note that, since DRNets are an unsupervised framework, we can apply the restart (Gomes et al. 1998) mechanism, i.e., we can re-run DRNets for unsolved instances. Specifically, since DRNets directly incorporate logical constraints, we can check whether those constraints are satisfied at the end of a run. If not, for instances with violated constraints, we re-run the algorithm again on them. We only applied restart mechanism on Siii--NNST--Suuokke and other NP-C problems (in the appendix) such as 3-SAT problems and standard Sudoku problems. For crystal-structure phase mapping, the results generated from one run of DRNets is already good enough.
A.4 EXPERIMENTAL CONFIGURATION
All the experiments are performed on NVIDIA Tesla V100 GPU with 16GB memory. For the training process of our DRNets, we select a learning rate in (0.0001, 0.0005, 0.001} with Adam optimizer (Kingma & Ba 2014), for all the experiments.
For baseline models, we followed their original configurations and further fine-tuned their hyper- parameters to saturate their performance on tasks.
A.4.1 MULTI-MNIST-SUDE
For Multi- MNIST-Sudoku, we compared DRNets with CapsuleNet (Sabour et al. 2017) and ResNet (He et al.) 2016). Because Sabour et al. (2017) did not provide a source code for CapsuleNet, we adopted the implementation Laodar (2017), with minor modifications. For ResNet, we adopted 18-layer ResNet architecture (Khanre 2017) to saturate its performance.
In Multi-MNISTT-uudnn optimization mode of DRNets, we generated 160, 000 input data points that all come from the test set of MNIST (LeCun et al.) 1998) and every 16 data points form a 4-by-4 overlapping Sudokus. Thus, these 160, 000 data points form 10, 000 Sudokus. These 10, 000 Sudokus are used as the test set and shared across DRNets and baselines. For the generalization mode of DRNets, we split the training set of MNIST into three parts: 160, 000 data points for DRNets learning, 23, 000 original MNIST images for training conditional GAN and another 160, 000 data points for validation. Note that these three datasets are disjoint. Baselines share the same training set as the generalization mode of DRNets. By using the constraint-awart SGD, DRNet batches every 16 data points together, which forms an overlapping Sudoku as well as a maximal connected component in the constraint graph, to enforce the All-Different constraints among the cells of each Sudoku.
a data point corresponds to a 32 x 32 image of overlapping digits. For the
DRNet for Multi-MNIST-Sudkku the encoder is made of two ResNet-18 models adapted from the PyTorch source code. The output layer for the first network has dimensions, which models the two distributions P; and Q, for the two overlapping digits. Another network outputs eight 100-dimensional (800 dimensions in total) latent encoding ži,j to encode the shape of the possible eight digits conditioned on the input mixture, and is used by the generative decoder to generate the reconstructed digits. We use a conditional GAN (Mirza Osindero 2014) as our generative decoder, which is pre-trained using the digits in the partial training set (see the paragraph above) of MNIST. Note that this is the only supervision we have in this task, which is even weaker than the general concept of the weakkyysuperviise
setting (Zhang et al.] (017). We adopted the implementation of Ninee (2019) for our conditional GAN. On the other hand, the 10,000 overlapping Sudokus in the test set were all generated using the digits in the fest set of MNIST, which had never been
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Accurucy (%)
Method
Time
Digit
Sudoku
DRNets (Optimization Restart)
100.0
100.0
50min
DRNets (Optimization)
99.9
98.6
28min
DRNets (Oplimization w/o Reasoning)
88.8
15.0
110min
Nets (Generralizatio)
98.0
75.7
13mio+4hrs
CapsuleNet
97.9
50.9
1min+30min
CapsuleNet local search
97.9
$7.8
3hrs+30mins
ResNet-18 18
97.7
68.5
3min+ 10hrs
ResNet-18 local search
97.7
88.3
3hrs+
Figure 9: Accuracy comparison. We show "test time + training time" for supervised baselines and the generalization mode of DRNet, and "solving time" for the optimization mode of DRNets.
seen, even by the conditional GAN. Moreover, we overlap the images of two digits pixel-wisely, maximizing the whiteness of the two images. For robustness concern, we used L1 loss as the reconstruction loss between the reconstructed mixture and the original input. For the initial weights, we set 0.01 for the cardinality constraints, 1.0 for the All-Different constraints, and 0.001 for the L1 loss. Finally, we trained DRNets for 100 epochs with a batch size 100, and took 50 minutes to finish the optimization and achieve the reported performance for the 10,000 overlapping Sudokus.
For the generalization mode of DRNets, we first "train" DRNets on the training set and validate its generalization performance the validation set to apply the early stop mechanism. Finally, we start from the "trained" DRNets and further optimize for 25 steps on the test set to achieve the reported performance. Note that, to generalize well on the test set, we "trained" DRNets for longer time than the optimization mode. Essentially, the procedure of the generalization mode of DRNets is similar to standard supervised learning process except that do not need labels supervise DeNess. In contrast, DRNets are really "self-supervised" (Jing & Tian 2019) by the Sudoku rules and the self-recontttuctiion instead of the standard supervision by labeled data. Note that, during the test, instead of predicting the overlapping digits directly as other networks, we further optimize DRNets on the test set for 25 epochs achieve better result.
A.4.2
PETRUCCTRE-PPAASE.E PHASE-MAPPINE
We illustrate DRNets for crystal structure phase mapping for two chemical systems: (1) ternary Al-Li-Fe oxide system (Le Bras et al., 2014), which is theoretically-bait with ground-truth solutions, and (2) ternary Bi-Cu-V oxide system, which is a more challenging real system obtained from chemical experiments and is more noisy and uncertain. For each system, the input data points are mixtures of XRDs, associated with a composition graph identifying elemental compositions and the constraint graph of data points. Specifically, each XRD data point associated with a 3-dimensional composition vector, which the proportion of the three different metal elements that data point (e.g.. [80% of A1, 5% of Fe, 15% of Li)) and could help identify possible phases. Then, we can locate each data point into a triangular system. Note that, since the vector is a probability distribution, there are only 2 degrees of freedom and can plot in 2-D triangle (See Fig IT}. After locating each data point into the 2-D triangle as vertices, we did a Delaunay triangulation over those points build edges among vertices. Therefore, we can use Breadth- First Search on this graph to sample paths in the composition graph and infer thermodynamic rules accordingly.
synthetically generated,
The XRD pattern of each data point a D-dimensional vector representing the intensity of the mixture of XRDs at different diffraction angles (referred as Q values). For AI-L.--e oxide system, we have 231 data points (mixtures of XRDs) in the composition graph, 159 stick patterns for the possible phases, and each data point has 650 different Q values Q, E [15", 80%] and the corresponding intensities I, E [0, 1). For Bi-Cu-V oxide system, we have 353 data points in the composition graph, 100 stick patterns for the possible phases, and each data point has 4096 different values Q, E [ 45% and the corresponding intensities I E [0. 1]. To better utilize the memory, we nlwn-ssamped the raw data of Bi-Cu-V oxide system to 512 different values. Note that, though have hundreds of possible
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Prototypes (ICDD stick patterns)
Input:
XRD patterns in the Composition Graph
Decomposed Phuse Putterns
Reconstructed
Structured Latent Space
XRD Patterns
Generative Decoder (GMM)
T heermodynamics Rules:
Encoder
Gibbs Phase Rule
batch
Reconstruction
Gible-Alloying Rule
Phase Field Connectivity
sampled path
Overall Objective
Multiplicative Shifting
Global Constraints
Butching
Controller
Weightt Controller
Reasoning
Prior lediee
Reasoning Modules
Loso
(b)
Deep reasoning networks (DRNets) for cystaal-ttucturre paruucture-phaeeeeee tatase-mapping (a) Prior knowledge includes the ICDD stick patterns of possible pure phases, which are used to build the GMM generative module the decoder, and the thermodynamic rules that help DRNets reason about mixture of XRD patterns. (b) reasoning modules batch data points involved in connected component
Figure 10:
of the constraint graph (a path in the composition graph) together, enforce that the structure of the latent space satisfies prior knowledge, and dynamically adjust the weights of constraints based their satisfiability. (c) The overall objective combines responses from the generative decoder and the reasoning modules.
(015 AL.5 Fe, 15% L.
Figure 11: The composition graph of the Al-Fe-Li oxide system. The red path is sampled path in the composition graph.
pure phases for each system, only a few phases would appear. For example, Al-Fe-Li oxide system, only 6 of them appear and there are 15 different mixtures of those 6 pure phases exist in this system. For the Bi-Cu-V-O system, there are 13 pure phases and 19 different mixtures. Note that, each XRD data point is like a cell in the Malti-MNIST-Sudooke (with mixed pure phases) and each pure phase is like digit. For Polti--NISTT-Suddkke we know priori that there are exact digits in each cell but the number mixed pure phases in each XRD undetermined to 3). Moreover, the number of possible candidate phases is way more than possible digits (e.g.. 159 vs 8), which is the reason why this task is so challenging.
We also collected a library of possible crystal structures from the International Centre for Diffraction Data (ICDD) database. Each crystal structure (also named phase) given as list of diffraction peak location-amplitud, pairs, (referred to as stick pattern), representing the ideal phase patterns measured in perfect condition (see Fig 12}. To model more realistic conditions, DRNets simulate the real phase patterns from stick patterns using Gaussian mixture models, where the relative peak locations and mixture coefficients are given by the stick locations and amplitudes. Moreover, the peak width, multiplicative location shift, and possible amplitude variance parameterized by the latent encoding ži,j and used by the generative decoder to generate the corresponding possible phase patterns in the reconstructed XRD measurement.
Imposing thermodynamic rules is challenging, especially when constraints, such as phase field connectivity and Gibbs-alloying rule, potentially concern all data points in the composition graph.
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L00
375
130
1.30
8.5
830
Figure 12:
Some examples of stick patterns and their corresponding Gaussian Mixture Models. The horizontal axis denotes the Q values, and the vertical axis denotes the diffraction intensity.
In Multi-MNIST-Sudkku where each overlapping Sudoku naturally forms the maximal connected components in the constraint graph, we easily batch every 16 data points together reason about the All-Different constraints among them. However, in Crystal Sitiial-ttrucuuu--Paa---e- Phase-Mapping, since the maximal connected component involves all data points in the composition graph, neither batching all data points into the memory nor reasoning about the whole graph is tractable. Therefore, we devised strategy of sampling the large connected component through many local structures (still connected components) and iteratively solve each of them. Specifically, for each oxide system, we sampled 100,000 paths in the composition graph via Breadth First Search to construct a path pool. Then, for every iteration, DRNets randomly sample path from the pool and batches the data points along that path (see [Ic Finally, we only reason about the thermodynamic rules along the path. By iteratively solving sampled local structures (paths) of the "large" maximal component, we can cost-efficiently approximate all global constraints.
We summarize the thermodynamic rules we imposed in DRNets:
Gibbs Phase Rule: This rule states the maximum number of co-cxisting phases, which is imposed via our relaxation of the k-sparsity constraints.
Gibbs-Aloyyng Rule: This rule states that if "alloying" happens, then the maximum number of possible co-existing phases should decrease by one. "Alloying" is phenomenon that the stick locations of a phase (crystal structure) shift (change) along with adjacent data points. DRNet explicitly models the shifting ratio in the generative decoder and penalize the difference between adjacent data points along our sampled path. The reasoning module keeps track of the difference of shifting ratio between adjacent data points, and when it is larger than threshold (0.001), ), we confirm the existence of "alloying" and reduce the maximum number of possible co-cxisting phases by one via adjusting the threshold e in the k-Sparsity Constraints.
This states that the distribution (also referred as activation) of a phase field should form a connected component the composition graph, and the variation of the activation of each phase should also be smooth (see Fig [3). (Herein, the phase field refers to the co-existence of a combination of phases, including the existence of a pure phase.) We impose this rule by penalizing the difference of the phase distribution P: between adjacent data points along our sampled path.
Phase Field Connectivity:
Multiplicative Shifting: This states how cubic crystal structure shifts when "alloying" happens, and this can also be used to approximate the shifting of other crystal structures. We explicitly modeled the multiplicative shifting in our generative decoder.
Noise Threshold:
To remove negligible activations that are mainly caused by noise we applied simple post- proccssinn that cuts-off all the activations that are lower than 1.0%.
Here, we visualized the DRNets' solution of Bi-Cu-V oxide system (see Fig 13 and the comparison among different methods Fig I4p.
In our comparison, evaluated the percentage of data points or phase field that satisfy each thermodynamic rule. Though IAFD enforced the thermodynamic rule using an external mixed-inte mixed-integer programming module, it may compromise some rules to achieve better reconstruction error, which explains IAFD's result for Bi-Cu-V oxide system. The phase fidelity loss we mentioned in our comparison is the JS distance between the de-mixed pure phase and the closest ideal phase generated
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Figure 13: DRNets' solution for the Bi-Cu-V oxide system. a. The de-mixed crystal phases for the 353 XRD measurements of the Bi-Cu-V oxide system (each plot includes the signal for the recognized phase and the corresponding ICDD stick pattern). b. DRNets' phase concentration maps for the corresponding phases the left of the map. Dot sizes proportional to their estimated phase concentrations and heatmap denotes estimated shifting (alloying). e. DRNets' crystal phase map for the Ei-Cu-V-Q system the composition graph; the phase fields are labeled with corresponding crystal phases.
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Figure 14: Comparison of the activation map and the heatmap of L1 reconstruction loss for different methods for the Bi-Cu-V oxide system: Each row denotes the activation of the different phases for the the different methods; Though we do not have ground truth for the Bi-Cu-V oxide sys- tem, the solution generated by DRNets satisfies all thermodynamic rules with excellent reconstruction performance; The heatmap on the right shows that DRNets reconstruct the XRD measurements much better than other methods with respect to the (log scale) reconstruction under physical constraints of decomposed phases; In addition, materials science experts thoroughly checked DRNets' solution of Bi-Cu-V oxide system, approved it, and subsequently discovered a new material that is important for solar fuels technology.
using the ICDD stick patterns and the physical model proposed in Le Bras et (2014). The reason of using JS distance to measure the fidelity is that the location of peaks are the most important characteristics of phase pattern. Therefore, by normalizing the XRD patterns of pure phases into probability distributions, we can use the JS distance to measure the mismatch of "peaks" between them.
In terms of the optimization process, DRNets took about 30 minutes to achieve the reported per- formance for both systems. IAFD and NMF-K have similar time performance but a much worse performance w.r.t. the solution quality. In fact, for the Bi-Cu-V oxide system, both NMF-k's solution and IAFD's solution are not physically meaningful.
In summary, by combining reasoning and deep learning, DRNets significantly outperformed the state of the art and human experts on the crystal-Structure Prructure-Phas--Mappee Phaping instances, recovering more precise, interpretable, and physically meaningful crystal structure pattern decompositions, and even solving phase diagrams of chemical systems that had not been solved before, such as the Bi-Cu-V-0 system, but also other systems not reported here.
A.4.3 OTHER EXPERIMENTS FOR COMBINATORIAL PROBLEMS
As a proof of concept how DRNets can encode general combinatorial constraints using our entropy- based continuous relaxation, we solved 9-by-9 Sudoku puzzles and Boolean satisfiability problems (SAT) using DRNets. For those two tasks, we use a 3-layer-fully-connees network as our encoder and the reasoning modules.
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Figure 15: standard 9-by-9 Sudoku puzzle: a partially filled Soduku has to be completed as valid Sudoku.
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For 9-by-9 Sudoku puzzles, we generated 10,000 instances using the dataset gathered by Gordon Royle (2014). where each Sudoku instance has 24 to 32 (uniformly distributed) known cells and is guaranteed to have one unique solution (e.g.. see Fig [ Because standard 9x9 Sudoku puzzle requires reasoning about the unknown structure based on given clues, we need to treat each entire Sudoku single input data point. Therefore, in this task, even the All-Different constraints conceptually the local constraints since each of them only concerns a single data point. We used a one-hot encoding for digits 1 to 9 and the empty cell (denoted as 0), and the entire Sudoku is an network with batch normalization (loffe& & Szegedy 2013) as the encoder, where every hidden layer has 2048 units and the output an 8 81-by-9 matrix, which represents the digit distributions (1 to 9) for 81 cells. Moreover, we enforced the distribution of every known cell to collapse to the digit in that cell. For the initial weights, we set 0.0001 for the cardinality constraints and 1.0 for the All-Different constraints. Finally, we trained DRNets for 800,000 iterations with a batch size of 500, and it took hour to solve the 10,000 9x9 Sudokus with the accuracy reported in this paper.
8 810-dimensional input data. We used a coayyeerfulll-oonnetete
In our experiments, DRNets achieved the same level of performance as the Recurrent Relational Networks (RRNets) (Palm et al., 2017,, which is the state-of-the of-the-artttttooo art supervised deep learning 9x9 Sudoku solver (see Table I.
Instances (10,000)
DRNett 81.0% (4min)
DRNctt Restart
NeuralSAT
PDP
RRNets NA
3-SAT n=30 m=
99.0% (33min)
45.5% (2min+1hr) Ihr)
78.9% (5min+2hr)
3-SAT n=30 m=2
63.3% (7miin)
94.0% (47min)
26.1% (3min+1hr) Ihr)
62.2% (8min+2hr)
NA
3-SAT 1=100 m=430
34.7% (17min)
77.9% (2hr)
4.7% (Smin+1ar)
31.4% (2hr+2hr))
3-SAT n=30, m=90
97, 9% (5min)
99.9% (6min)
78.5% (2min + Ihr)
99.1% (4min + 2hr)
NA
3-SAT n=50, m= 130
98.2% (Tauin)
99.4 % (8min)
70.1% (3min + Iha)
99.2% (7min + 2hrr)
NA
3-SAU n= n=100, m=300
98.1% (20min)
99.7% (22min)
52.9% (5min + Ihr)
99.1% (2ar + 2hr)
Sudoku
59.5% (Ihz)
99.8% (Ihr)
NA
NA
99.6% (Imin+ Iday)
Table I: Percentage of instances solved for 3-SAT (m/n = 4.3 and m/n = 3.0) and standard 9x9 Sudoku (24 to 32 known cells). We show the "test time training time" for supervised baselines and the "solving time" for our unsupervised DRNets. The units min, hr, day denote minute(s), hour(s) and day(s). m, denote the number of literals and clauses, respectively. NA, not applicable. DRNets, without supervision, outperform the supervised state of the art.
For SAT problems, we generated 10,000 satisfiable random 3-SAT instances of different difficulties based the number of literals and the number of clauses m. and our goal is to find a valid assignment for each literal. We challenged our DRNet with the hardest random 3-SAT instances, where #clauses/Hiierras=44.3 (Mitchell et al.) 1992). i.e., n = 30, m = 129, 50, m = 225 and 100, m = 430. For easier instances (e.g. #clauses/fliteraass = 3.0), DRNets can almost solve all instances (see T..
We use a fanyerr-ully-conneette network as the encoder, where the number of hidden units in the network is 2048, 2048, 2048. We used the standard CNF representation of 3-SAT as the input data, so that each data point is an m-by-3 matrix and the three values in the j-th row represent the three literals in the j-th clauses. For the initial weights, we select a value from {0.05, 0.03, 0.025, 0.02, 0.01 to be the weight of the entropy loss we described in the Fig.4 of the main paper. For the three settings of different difficulty, we consistently trained DRNets with a batch size of 100 and the running time for solving 10,000 instances varies from several minutes to couple of hours.
We compared DRNets with Neuro Selsam et al. (2018) and PDP (Amizadeh et al. 2019). Both Ne and PDP are the state-of-the tith-therr art deep learning SAT solvers with one-bit supervision. In addition, PDP needs extra optimizing process to solve SAT instances during the test phase, where it also applied the restart mechanism in their framework. For fair comparison, we saturated the performance of all our baseline models. For all instances, DRNets took less than 2 hours to achieve the reported performance with the restart mechanism. Without supervision, DRNets outperformed both supervised baseline models.
Interestingly, though DRNets are best suited for problems that combine deep learning and reasoning, such as di-mixing promising result in pure combinatorial problems. These results further demonstrate that DRNets can encode a broad range of combinatorial constraints and prior knowledge and effectively combine deep learning with reasoning.
or crystal structure phase mapping, it still achieved such a
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Reviews:
[{"review_id": "rkeTAUPAYH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a new encoder-decoder framework that combines prior knowledge-based regularization and constrained reconstruction for unsupervised and weakly-supervised classification in structure rich scenarios. This framework injects prior knowledge in the form of relaxed constraints that act as regularization during the training of the encoder network. Some of the constraints concern sets of training examples. In this case, the paper proposes corresponding sampling schemes. Three experiments demonstrate the efficacy of the model. The first is a synthetically created 4x4 Sudoku made of overlaid MNIST digits. The other two are based on predicting crystal structures from x-ray diffraction measurements. Here, the first experiment is on simulated data for the Al-Li-Fe oxide system, while the other is performed on real measurements for the Bi-Cu-V oxide system.\n\nOverall, I believe that the proposed framework could be a significant contribution to the fields of representation learning and constrained optimization. However, the paper exhibits serious shortcomings, which require revision. \n\nFirst, the positive aspects of the paper:\n•\tThe framework is simple yet ingenious. It makes intelligent use of constraints in the form of regularization to guide the training of the encoder. Furthermore, it enables the direct design of the latent representation through the use of (pre-trained) generative models for constrained reconstruction of data points.\n•\tThe proposed entropy-based method for relaxation of discrete constraints is intuitive and potentially adaptable for further constraints.\n•\tThe experiments presented in this paper are well chosen. They demonstrate the contribution of the model to both general CV data as well as a specialized domain, where it can solve both simulated and real scenarios.\n•\tThe paper provides an extensive literature survey, which makes it easy to embed the presented work in the proper context. However, I propose to remove the paragraph titled “Other less closely related work” as the connection to the current work is not clear, and the space could be used more effectively (see below).\n\nUnfortunately, this paper has a couple of major flaws:\n•\tThe results for DRNets (Generalization) on the MNIST Sudoku are compromised because the model trained on the test set for 25 epochs after being trained on the training set. Honestly, I was baffled to read the following sentence in the appendix: “Note that, during the test, instead of predicting the overlapping digits directly as other networks, we further optimize DRNets on the test set for 25 epochs to achieve a better result.” What is more, the main paper does not even mention this fact!\n•\tAlthough this paper relies on empirical verification of its proposition, the experimental results are almost impossible to interpret with just the information provided in the paper. Both experiments are poorly described, and even after reading the appendix several times, some serious detective work was necessary to piece together what happened in the experiments. The XRD experiments are especially hard to decipher, even with a physics background. Many vital components remain shrouded in mystery: What is a composition graph, and how are the paths sampled from it? How does the restart method, which is part of the results, work? Why are only six phases shown in the phase concentration visualizations if there were 159 possible phases? Are these the first six, a random subset, or were the other phases not realized?\n•\tThe paper introduces the constraint-aware SGD algorithm to incorporate batching rules into the training of the encoder. On pages 2 and 6 and in Algorithm 1, I found the statement that the weights for each constrained are updated dynamically. However, that is where the information on the dynamic update method ends. Nowhere in the paper or the appendix did I find an explanation of how this is done. As this mechanism is a critical component of the proposed framework, the absence of an explanation is a significant oversight.\nOther remarks:\n•\tIn the context of global constraints, the paper talks about a constraint graph. If I understand the creation of this graph correctly, this graph has several connected components in which every element connects to every other element. As such, this seems to be a collection of sets rather than a real graph. This is especially confusing in the case of XRD, where all data points are in the same global constraint, leading to a fully connected “graph”.\n•\tAlthough I appreciate the reference to Kahneman’s model of the mind, I suggest to remove the first two paragraphs from the introduction and use the space to motivate the de-mixing problem instead. While it is a compelling (but not novel) observation, the analogy to system 1 and system 2 does not benefit the proposed work in the slightest.\n•\tIn general, I fail to see the connection between reasoning and the proposed work. The model itself is an encoder-decoder network that cannot reason. It does not discover any new rules during training. All the reasoning has to be done manually beforehand to be then incorporated in the form of constraints. To clarify, I do believe that there is value in the presented work, but not necessarily in the way, it is advertised.\n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "rJeunhge9r", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This work proposes a framework for solving de-mixing problems. The hard constraints from human inputs about a specific problem are relaxed into continuous constraints (the \"slow\" reasoning part), and a reconstruction loss measures the fitness of the inferred labels with the observations (the \"fast\" pattern recognition part). Due to the relaxation inference becomes an optimization problem, and on a Sudoku task and a crystal-structure-phase-mapping recovery task (both de-mixing tasks), the proposed method gets very good performance (100% for all Sudoku tasks including one in the appendix).\n\nPros:\n1. The method works well for the two demixing tasks.\n2. It \"led to the discovery of a new material that is important for solar fuels technology\"\n\nCons:\n1. The generative decoder seems to be pretrained on both tasks instead of learned (correct me if I misunderstood), and I'm not sure if this approach can work in cases where we don't have access to such a generative decoder, so branding the approach \"deep reasoning network\" might be an overclaim.\n2. No reasonable baselines are used: The supervised baseline in Sudoku does not use those handcrafted constraints at all. Given pretrained decoders, a reasonable baseline would be randomized optimization methods such as simulated annealing, which might also solve the two tasks listed here.\n3. This paper proposes a deep reasoning framework with relaxation and continuous optimization, but it is unclear whether this can solve general reasoning problems such as multi-hop QA or some NP-hard integer programming problems.\n\nQuestions:\n1. In algorithm 1, how are the penalty weights and thresholds adjusted?\n2. How to determine whether a run needs to restart?\n\nOverall this work points an interesting direction of combining reasoning and pattern recognition in the same network and the proposal works well on two de-mixing problems. However, I am not convinced that the proposed solution can generalize to tasks other than the tasks proposed here, and the usage of pretrained generative decoders undermines the significance of this work. Therefore, I am inclined to reject this paper.\n\n\n---updates after reading authors' rebuttal----\nThanks for revising the paper and addressing my concerns! However, my concern Con #2 has not been fully addressed. I think a reasonable baseline (at least for Sudoku) is simulated annealing, such as in https://www.researchgate.net/publication/220704743_Sudoku_Using_Parallel_Simulated_Annealing. I believe that with restarts those baselines would also solve the Sudoku problem.\n\nAnother concern I still have is the claim of \"reasoning\", and I'd suggest to narrow down the claim to be only on pattern de-mixing, since the reasoning part seems to be writing down continuous constraints from the discrete constraints (same as the concern in review #3). Although the proposed approach can solve some NP-C integer programming problems, it is unclear based on the experiments here whether it can work for general reasoning tasks (e.g., DROP https://allennlp.org/drop or listops https://arxiv.org/pdf/1804.06028.pdf) without writing new rules manually.\n\nBesides, after reading Reviewer 3's comments, I also feel it unsuitable to train DRNet (generalization) on test set for 25 epochs even though you made it explicit in the revised paper. I'd recommend removing that experiment since it doesn't change this work that much.\n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "ByxENYaL9r", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper introduces a deep reasoning networks for de-mixing overlapping patterns with some logic constraints. There are two applications considered in the paper: de-mixing overlapping hand-written digits and inferring crystal structures of materials from X-ray diffraction data. The experiments indicate the proposed method work pretty well on these tasks.\n\nI like the general idea of this paper, since it has the flavor of combining deep learning with logic rules, although I feel weird to view the generative decoder as thinking fast and the reasoning modules as thinking slow. The notion of thinking fast and slow in the model does not well match the intuition given in the first paragraph of the introduction. The so-called reasoning module is essentially some contraints (i.e., regualrization losses) and a training data sampler. It is far away from the concept of (symbolic or logic) reasoning. There is not too much reasoning happening here. The way the paper relaxes the discrete logic constraints to continuous and differentiable objective that can be jointly optimized by SGD is interesting, which is similar to [Harnessing deep neural networks with logic rules, ACL 2016]. The carefully designed training data sampler that samples data according to a constraint graph also resembles GraphRNN, as the authors have mentioned in the paper. I feel the combination of these techniques is definitely interesting but also somehow incremental. I am not a big fan of some big claims in the paper. The reasoning modules are not what I expect.\n\nFor the experiments, I think the authors do a good job presenting these experimental details and evaluations. These experiments are interesting and also show some advantages of the propose method. However, some baselines are also doing pretty well, indicating that the task is not difficult in general.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper received mixed reviews of WR (R1), WR (R2) and WA (R3). AC has carefully read all the reviews/rebuttal/comments and examined the paper. AC agrees with R1 and R2's concerns, specifically around overclaiming around reasoning. Also AC was unnerved, as was R2 and R3, by the notion of continuing to train on the test set (and found the rebuttal unconvincing on this point). Overall, the AC feels this paper cannot be accepted. The authors should remove the unsupported/overly bold claims in their paper and incorporate the constructive suggestions from the reviewers in a revised version of the paper.
Decision: Reject