Monotonic neural network: combining deep learning with domain knowledge for chiller plants energy optimization
Authors: Fanhe Ma
Faen Zhang
Shenglan Ben
Shuxin Qin
pengcheng Zhou
Changsheng Zhou
Fengyi Xu
Venue:  (2021)

Abstract: In this paper, we are interested in building a domain knowledge based deep learning framework to solve the chiller plants energy optimization problems. Compared to the hotspot applications of deep learning (e.g. image classification and NLP), it is difficult to collect enormous data for deep network training in real-world physical systems. Most existing methods reduce the complex systems into linear model to facilitate the training on small samples. To tackle the small sample size problem, this paper considers domain knowledge in the structure and loss design of deep network to build a nonlinear model with lower redundancy function space. Specifically, the energy consumption estimation of most chillers can be physically viewed as an input-output monotonic problem. Thus, we can design a Neural Network with monotonic constraints to mimic the physical behavior of the system. We verify the proposed method in a cooling system of a data center, experimental results show the superiority of our framework in energy optimization compared to the existing ones.

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MONOTONIC NEURAL NETWORK: COMBINING DEEP LEARNING WITH DOMAIN KNOWLEDGE FOR CHILLER PLANTS ENERGY OPTIMIZATION
Anonymous authors Paper under double-blind review
ABSTRACT
In this paper, interested in building domain knowledge based deep learn- ing framework to solve the chiller plants energy optimization problems. Com- pared the hotspot applications of deep learning (e.g. image classification and NLP), it is difficult to collect enormous data for deep network training in real- world physical systems. Most existing methods reduce the complex systems into linear model to facilitate the training on small samples. To tackle the small sample size problem, this paper considers domain knowledge in the structure and loss de- sign of deep network to build a nonlinear model with lower redundancy function space, Specifically, the energy consumption estimation of most chillers can be physically viewed Neural Network with monotonic constraints to mimic the physical behavior of the system. We verify the proposed method in a cooling system of data center, perimental results show the superiority of our framework in energy optimization compared to the existing ones.
input-output monotonic problem. Thus, we can design a
INTRODUCTION
The demand for cooling in data centers, factories, malls, railway stations, airports and other build- ings is rapidly increasing, as the global economy develops and the level of informatization improves. According to statistics from the International Energy Agency (TEA 2018), cooling energy consump- tion accounts for 20 of the total electricity used in buildings around the world today. Therefore, is necessary to perform refined management of the cooling system to reduce energy consumption and improve energy utilization. Chiller plants cooling system. Due to the non-linear relationship between parameters and energy consumption, and performance changes due to time or age, deep learning is very suitable for modeling chiller plants.
of the main energy cerggy-consuming equipment of the
In recent years, deep learning (Goodfellow et al., 2016) research has made considerable progress, and algorithms have achieved impressive performance on tasks such as vision (Krizhevsky et al.) He et al. 2016). language (Mikolov et al. 2011 Devlin et al. 2018), and speech (Hinton et al. 2012 Oord et al. 2016), etc. Generally, their success relies on a large amount of labeled data, real-world physical systems will make data collection limited, expensive, and low low-quality due to security constraints, collection costs, and potential failures. Therefore, deep learning applications are extremely difficult to be deployed real-world systems.
2012
There are some researches about few sample learning summarized from Lu et al. (020 which focusing on how to apply the knowledge learned in other tasks to few sample tasks, and applica- tions in computer vision, natural language processing, speech and other tasks. Domain Knowledge that has been scientifically demonstrated, however, is more important in few sample learning tasks, especially in the application of physical system optimization.
Domain knowledge can provide more derivable and demonstrable information, which is very helpful for physical system optimization tasks that lack samples. We discussed the method of machine learn ing algorithms combined with domain knowledge and its application in chiller energy optimization in this article.

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In partieular, we propose a monotonic neural network (MNN), which can constrain the input-output of the chiller power model to conform to physical laws and provide accurate function space about chiller plants. Using MNN for system identification can help the subsequent optimization step and improve 1.5% the performance of optimization compared with the state-of-the-art methods.
2 BACKGROUND AND RELATED WORK
Chiller plant[] energy optimization is an optimization problem of minimizing energy. In order to simplify the optimization process, the optimized system is usually assumed to be stable, which means that for each input of the system, the corresponding output is assumed be time-independent
Mostly used methods are model-based optimization (MBCF) (Ma & Wang 2009; Ma et al. 2011
Huang & Zuo| 2014). Although Some research using Reinforcement learning model for optimal control (Wei et al. 2017 Li et al. 2019 Ahn & Park 2020). However, applying RL to the con- trol of weal-world physical systems will be caused by unexpected events, safety constraints, limited observations, and potentially expensive or even catastrophic failures Becomes complicated (Lazic
et al. 2018
MBO has been proven be feasible method to improve the operating efficiency of chillers, which uses chiller plants model to estimate the energy consumption with given control parameters under the predicted or measured cooling load and outside weather conditions. The optimization algorithm is then used to get the best value of the control parameter to minimize energy consumption (Malara al. 2015). The model can be a sryysics-based model or a machine learning model. Physics-bai -ased models are at the heart of today's engineering and science, however, it is hard to apply due to the complexity of the cooling system. Experts need to spend a lot of time modeling based on domain knowledge (Ma et al. 2008). When the system changes (structure adjustment, equipment aging, replacement), it needs to be re-adapted. In recent years, the data-driven method has gradually be- come an optional solution. Its advantage lies in the self-learming ability based historical data and the ability to adapt to changes. Thanks to its stability and efficiency, linear regression is the mostly used modeling method in real-world cooling system optimal control tasks (Zhang al. 2011 Lazic jet al. 2018). But ordinary linear models cannot capture nonlinear relationships between parameters and energy consumption, and polynomial regression is very easy to overfit. With the remarkable progress of deep learning research, some studies apply it in cooling system (Gao 2014 Evans & Gao 2016 Malara et al. 2015). Deep learning is very good at nonlinear relationship fitting, but relies on a large amount of data and is highly nonlinear, which brings great difficulties to subsequent -ecision-making. Due the inability to obtain a large amount of data, frontier studies have be- gun to consider the integration of domain knowledge into the progress of system identification and optimization (Vu et al.) 2017] Karpatne et al., 2017 Muralidhar al.) 2018 Jia et al.) (020). The combination methods made laudable progress, although it is still a relatively early stage.
In conclusion, reinforcement learning approach either requires a detailed system model for simula- tion an actual system that can be tested repeatedly. The cooling system is too complex to simulate, the former is impossible. While in actual system design and implementation, the latter may be im- practical. The MBO method has been proven to be feasible in optimal control, and the optimization performance is determined by the system identification model. However, physical model is complex and time-consuming, linear model in the machine learning model has poor fitting ability, neural network requires large scale datasets, and its highly nonlinearity is not conducive to subsequent optimization step. Domain knowledge can provide more knowledge for machine learning, in this article, we make a theoretical analysis and methodological description about the combination of do- main knowledge and deep networks. In particular, we propose a monotonic neural network, which can capture operation logic of chiller. Compared with the above state of art method, MNN reduces the dependence on amount of data, provides more accurate function space, facilitates subsequent optimization steps and improves optimization performance.
How chiller plants work can see in appendix A.I
How MBO methods work can see in appendix A.3
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3 MACHINE LEARNING COMBINE WITH DOMAIN KNOWLEDGE
Consider general machine learning problem, let us explain the method of machine learning from another angle. It is well known that the life cycle machine learning modeling includes three key elements: Data, Model, and Optimal Algorithm.
f arg min Reep s.t. constraints FE7
First, function representation set is generated through model. Then Under the information con- straints of training datasets, the optimal function approximation is found in the function set through optimization strategies. Deep learning models have strong representation capabilities and huge function space, which is doublc-edged sword. In the case of few sample learning tasks, if we can use domain knowledge to give more precise function space, more clever optimization strategies, and more information injected into the training datasets. Then the function approximation to be solved will have higher accuracy and lower generalization error.
Prior knowledge is relatively abstract and can be roughly summarized as: properties (Relational, range), Logical (constraints), Scientific (physical model, mathematical equation). Several methods of how domain knowledge can help machine learning are summarized in this paper, as follows:
Scientific provides an accurate collection of functions. the physical model known but the parameters are unknown, machine learning parameter optimization algorithms and training samples can be used to optimally estimate the parameters of the physical model. This reduce the difficulty of modeling physical models.
Incorporating Prior Domain Knowledge into data. The machine learning algorithm learns from data, adding additional properties domain knowledge to the data will increase the upper limit of model performance, such as: Constructing features based on the correlation between properties; processing exceptions based on the legal range of properties; Data enrichment within the security of the system, etc.
Incorporating Prior Domain Knowledge into optimal algorithm. The optimization goals in ma- chine learning can be constructed according to performance targets. Therefore, logic constraints in be added penalty
domain knowledge that have an important impact on model performance to the optimization objective function. That will make the input and output of the model conform to the laws of physics, and improve the usability of the model in optimization tasks.
Incorporating Prior Domain Knowledge into model. Another powerful aspect of deep learning is its flexible model construction capabilities. Using feature ranges and logical constraints of domain knowledge can guide the design of deep learning model structure, which can significantly reduce the search space of function structure and parameters, improve the usability of the model.
4 CHILLER PLANTS ENERGY OPTIMIZATION
This section will introduce the application of using the machine learning combine with domain knowledge to optimize the energy consumption of chiller plants. The algorithm model mentioned below has been actually applied optimization method to optimize chiller plants. The first step is to identify the chiller plants. We decompose the chiller plants into three type models: cooling/chilled water pump power model, cooling tower power model, and chiller power model, see Equation 2
cooling system of a real data center. We use model-based
y PcH Pet PoowP РOнWР
4.1 MODEL WITH SCIENTIFIC
For the modeling of the cooling tower power and the cooling/chilled pump power, we know the physical model according to domain knowledge, that is, the input frequency and output power are cubic relationship (Dayarathna et al. 2015). see Equation
y = f(x; 0) Pas 10g (x/ Fan)3 02 (x=) + 01 (2r Fac)) + 0n]
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Where is the imput parameter: equipment operating frequency; Pde is the rated power. Fde is the rated frequency, which is a known parameter that needs to be obtained advance. 8,. 01, 82. Bg is the model parameter that needs to be learned.
4.2 FEATURES WITH PROPERTIES
For the modeling of chiller power, we can integrate the relationship information between properties into the features improve the fitting ability of the model by analyzing how the chiller plants work
in appendix A.1
UCH OK T Teandenser, Qeooling loads
(4a)
T Tondensser O Toow iny Feow pump
(4b)
Teow in X T Toow Tubs F/ T Teow_out O T Teondenser; Temm jnn: 1/ Pow.ppump Q cooling O (Tehw-in T put)) @chilleed.water_foo
(4c)
(4d)
(4e)
Chhilled water- X Foow.ppms
(4f)
See Equation4 lists the causal relations between yCH and the variables on the cooling side and chilled side, and the correlation between variables. Because Teow in and Teow out is autoregres- sive attribute related to time series, so cannot used as a feature. We will get features, list in Equation
XCH = (Twb; Tehor_outt Teho iny Feow pumpp Frans Fehw pump))
(5)
4.3 OBJECTIVE FUNCTION WITH LOGIC
For the modeling of chiller power, we choose to use MLP as the power estimation model of chiller in the choice model structure. The MLP model has the advantage to fit well on the nonlinear relation- ship between input and output. However, the estimated hyperplane of chiller power(c, fehater (x)) has the bad characteristics of non-smooth and non-convex due to limited data and the highly non- linearity of the neural network, resulting in the estimation hyperplane of total power, that will be optimized, (c, (x)) has multiple local minimum points, see figure 4.1 Moreover, the input and output of the model do match the operating principle of the chiller from the performance curve. This brings great difficulties to the optimization steps later, which is why deep learning is rarely used in the control of real physical systems.
(a) bad identification hyperplane.
(b) bad optimization hyperplane.
Figure 4.1: natural curve.
The chiller plants have the following operating logic, such as the cooling tower fan increases the fre- quency, and will decrease the power of the chiller, etc. So the model's natural curve parameters should be monotonous, see Table The natural curve output by the vanilla MLP model does not conform this rule, Figure 4.2
"hhe natural called sensitivity curve: the change curve of along certain dimension of X.
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Table 1: x Peи monotonicity
Monotonicity
F Fjan
Decrease
Feowspump Twb Fehw pump
Decrease
Increase
Increase
Tehw Tch
Decrease
Increase
[oowwpp poowpparppp
325
240
300
775
230
180
260
160
172
400 02. 160 47.
10.0
12.5
15.0
500
(a) bad natural curve of Fan..
(b) bad natural curve of F Feom puomp.
Figure 4.2: bad natural curve. Each curve a sample
Adding penalty for the inconsistency of the physical law (monotonicity) to the loss function can achieve the effect of incorporating the constraints of the chiller operating logic into the chiller model. Here we design two pairwise rank loss" for that:
LosS(yA: rank = CrossEntropy (Sigmoid(y vn). > un))
(6)
Loss(y's ()A (BA rank = max VA yBB) T[yA < yB)) + mar(0, yB - y/a) T(ya yB))
(7)
In Equation 6 we the sigmoid function to map the difference between the power estimated label of the A sample and B sample into the probability estimate of yA yB, and then use cross entropy calculate the distance between the estimated probability distribution Sigmoid(zj yB) and the true probability distribution I(y > yB) as a penalty term.
In Equation 7
when the estimated order of the label of sample and B sample does not match the truth, we use the difference of the estimated label of the label of A sample and B sample as a penalty.
Based on the addition of the above penalty items, the learning of the model can be constrained by physical laws, so that the natural curve of the model conforms to monotonicity, the effeet See Figure 4.3 and the estimated hyperplane is very smooth, and the optimized plane is also convex It easy to use the convex optimization method to obtain the optimal control parameters. see Figure 4.4
f,ttn sensitiity
L_oo_puump sensitiitt
240
260
228
200
3100
20.5
75.0
17.5
1000
125
15.3
05.
510
30.0
12.1
20.0
37.5
10.3
12.5
45.0
175
500
(a) good natural curve of Fjan.
(b) good natural curve of Feow.pump.
Figure 4.3: good natural curve.
47 is Indicator Function
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(a) good identification.
(b) good optimization.
Figure 4.4: good identification and optimization hyperplane.
The addition of the rank loss requires us to construct pairwise samples [( BB), 1(yA > yB)]. Part of the construction comes from original samples, and others need to be generated extra. First æg is copy from AA then æ selects a monotonic feature C. plus a small random value. Based on the order of B, and AA,, referring to the monotonicity of æ. will get the true power consumption comparison (y yB).
4.4 MODEL STRUCTURE WITH LOGIC
The former Section 4.3 describes the integration of logic constraints by adding penalty items to the loss function, so that trained model conforms to the physical of monotonic input and output. This section will describe how to use parameter constraints constrainls(0) and model structure design f to further improve the model's compliance with physical laws and model accuracy. Equation
y = constrainnst (6)), x-y satisfies Physical Law
Inspired by ICNN(Amos et al. (017). we designed Monotonicity Neural Network, which gives the model the monotonicity of input and output through parameter constraints and model structure design, called hard-MNN. Corresponding monotonicity through the objective and loss function called soft-MNN.
this is the model in the previous section that learns
4.4.1 IIARD-MNN
Model structure see Figure
4.5
The main structure of the model is a multi-layer fully connected
W
W
W
W
W
X'
Zn
Z1
Za
Zu-1
6
Z
y
W
W
W.r
Figure 4.5: hard-MNN. X is Input, y is Output, M mask layer, Z, hidden layer, W weights: W is passthrough layer weights, W is main hidden layer weights. W is output layer weights, activate function, is aggregation function.
feedforward neural network, and the mask layer function is added after the input layer identify

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the monotonic direction of 24. If z; decreases monotonously, take the opposite number, otherwise it remains unchanged.
if Increase fm(x)= f-x x if E Increase set
In the model definition, we constrain the weight to be non-negative (W 2 0, W 2 W 0). Combined with the mask layer, we can guarantee the physical laws of monotonically increasing or decreasing from the input the output. Because the non-negative constraints on the weights are detrimental to the model fitting ability, a "pass-through" layer that connects the input layer to the hidden layer is added to the network structure to achieve better representation capabilities. There are generally two ways of aggregate function, plus or concate, which can be selected as appropriate, the experimental results show that there is no significant difference.
W 2i- + W (=)
x
plus
2; = (
(10)
21-1; W x(]
concate
Similar to common ones are residual networks (He et al. 2016) and densely connected convolutional networks (Huang et al.) 2017). the difference is that they are connections between hidden layers. What needs to be considered is that the non-negative constraint of weights also detrimental to the fitting ability of nonlinearity. It makes the model only have the fitting ability of exponential low-order monotonic functions. Therefore, some improvements have been made in the design the activation function. Part of the physical system is exponential monotonic function, but in order to improve the versatility of the model, we designed a parametric truncated rectified Linear Unit (PTRelu which can improve the ability fit higher-order monotonic functions.
(x) min(a sigmoid(3xr) (0. a:))
(11)
0 are hyperparameter or as learnable parameters, a is the upper bound value of the output of the activation function, and / determines the smoothness of the upper bound to ensure its high-order nonlinearity and weaken the gradient explosion. Input-output comparison of activation function see Figure 4.6
5.
PTRelu/a=5)
ptrelu
ptrelutB=005) 0trelu(B=0071
5
1b
10
(a) PTRclu vs Relu
(b) PTRclu vs PTRelu
Figure 4.6: PTRelu.
In addition, extend the monotonic neural network to make it more general refer to (Amos al., 2017 Chen et al.) 2019). Such as: partial monotonicity neural network in A.4 monotonicity recur- rent neural network in A.5..eetc.
Adding each power model will get a total power model with convex properties, which is similar to ICNN. However, ICNN only guarantees the convex function properties of the objective function, which can facilitate the optimization solution but does not guarantee the compliance of the physical laws, nor the accuracy of the optimal value.

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5 EXPERIMENTS
We evaluate the performance MNN-based and MLPP-based optimization methods in a large data center cooling system. Since the performance of MBO mainly depends on the quality of the basic model, we first compare the accuracy of the two system identification models. Then we compare the energy consumption of the two models under the same cooling load and external conditions.
Comparison model estimation accuracy. From figure 5.1 we can know, the accuracy and stabil- ity of MNNs is better than MLP, because MNN provides a priori and more accurate function space.
MAPE
MLP
hard-MNN
Soft-MNN
Figure 5.1: Boxplot of mape of MLP, hard-MNN and soft-MNN, which trained on real data collected from cooling system of DC. Each model has the same number of hidden layers and the number of neurons in each layer, as well as the same training set, test set, and features. The result obtained after 100 non-repetitive tests.
Comparison of energy consumption. Considering that energy consumption not only related to interInal control but also related to the external conditions (cooling load and outside weather), in order to ensure the rationality of the comparison, we make PUE comparisons at the same wet bulb temperature. As shown in figure 5.2) hard-MNN more energy-efficient, stable and finally reduces the average PUE by about 1.5% than MLP.
NI2 winh FD hard Vay
3.54
3.58
2.55
3.54
3.50
1.50
3.55
22
22
23
24
26
Wet-Bulb Temperature
Figure 5.2: Energy consumption comparsion in real system. MLP is hard to be used in real world system optimization due to highly nonlinear, use mlp with local PID for safe constraints.
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A APPENDIX
A.1 COOLING SYSTEM
As shown in Figure A.I chiller plants are the main equipment of the cooling system. The chiller is used produce chilled water. The chilled water pump drives the chilled water flow the water pipe and distributes it to the air handling units (AHUs). The fan of AHUs drives the cold air to exchange heat with the indoor hot air for cooling rooms. this process, the heat obtained by the chiller from the chilled water needs to be dissipated into the air through equipment such cooling towers. Most of the heat exchange process uses water as a medium, and the equipment that drives the flow of the medium is cooling water pump. Chillers, water pumps, fans of AHUs and fans of cooling towers constitute the main components of the energy consumption of the cooling system. For more details, please refer to Stanford IIII (011)
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Chiller Plants
Cooling Water Pump
AHUs
Cooling
hot air
Tower
Chiller
cold air
Chilled water pump
Figure A.I: Cooling System Structure.
Table 2: Table of notations
Symbol
Description
Control vector- variables
State vector-variable
X
Features model, contains с and s Totoal power of chillers, cooling towers and water pumps Parameters of identification model Frequency of cooling water pump
y
peomppump
F fan
Frequency of cooling tower fan
Tub
Temperature of Wet bulb
Tewe out
Temperature of chilled water flow out chillers Temperature of chilled water flow in chillers Frequency of cooling water pump Temperature of cooling water flow out chillers Temperature of cooling water flow in chillers power chillers
Tohw in
Eehw-pump
Toowsnt
Teowsn
Pсн
Pet
power of cooling towers
Poowr
power cooling water pumps
POнWP
power of chilled water pumps
A.2 NOTATION
We have summarized the symbols used in the article, see T There are two types of variables for data collection in the cooling system: control variables and state variables and powers. Control variables are parameters that can be manually adjusted, state variables are factors that are subject to manual adjustment, but they all affect the energy consumption of the system. x is the input feature of models and is the output target of models. O represents the parameters of models. The symbols below represent actual variables in the cooling system. Feomspumpp fan are the control variables we want to optimize. Tun. Tehwomts Tetw Feto pumpp. T Toom_outs T Trom.n are environment v Рси, Pet, Pcowr, Poнwr are the power of each equipment in chiller plants.
A.3 OPTIMAL CONTROL
Chiller plants energy optimization is an optimization problem of minimizing energy. In order to simplify the optimization process, the optimized system usually assumed to be stable, which means that for each input of the system, the corresponding output is assumed be time-independent. Commonly used methods are model-free strategy optimization or model-based optimization. The strategy optimization method is to control according to the rules summarized by experience. The
TTeku out, Fehe pump also be controlled, but they will affect the energy consumption of AHUs. So in order simplify the optimization process, optimization control is performed them.

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model-based optimization method has two steps, including system identification and optimization, see Figure A.2
Chiller plants
Identificttio
state
Optimization
Model
Figure A.2: Mobel based optimal control. Solid line is identification step, dotted line optimization step.
The first step is to model the system, that is, building mapping function f x y between features and energy consumption as shown in Equaltion [2| this step is usually done offline. In the second step, a constrained objective function is created based the function of the first step, and then use the optimization algorithm to find the optimal value of the control parameter: The solved values will be sent the controller of the cooling system, this step usually performed online.
Lidenti cedantification.
y = f(x;
2.optmiization:
(12)
x" = arg min f(x; 0), s.t. some constraints
xEX
The modeling in the first step is the key step and the core content of this article, because it directly determines whether the implementation of optimization is troublesome, and indirectly determines the accuracy of the optimal value.
A.4 PARTIAL-MNN
Of course, when applied to other scenarios, the structure of hard-MNN not applicable because the features may not conform to all x-y monotonicity, so we expand hard-mma mnn to partial-mnn and model structure see Figure A.3 The PaN has one more branch network parts compared with hard-MNN, and the mask layer has also been modified.
The partial mask layer, see Equation [3 is designed to identify monotonic decreasing, monotonic increasing and non-monotonic features.
non-Monotonic
Jma (x)
Decrease
(13a)
x
Increase
fm2(x) = {fo
fmn (x) = 0
(13b)
fmi (x) + 0
The monotonic feature is input into the backbone network through the mapping of fmi of the mask layer, X,, = fmi (x). Non-monotonic features are input into the branch network x,, = (f (x)) through the fm mapping of the mask layer.
The branch network has no parameter constraints, uses the ordinary relu activation function, and merges with the backbone network each layer, see Figure A.3
12

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Under review as a conference paper at ICLR 2021
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Figure A.3: partial-MNN.
A.5 MRNN
MRNN replaces the main structure with RNN to support the modeling of timing-depende systems, and increases the monotonicity of the timing dimension by constraining parameters compared to MNN. As we mentioned carlier, the cooling system is dynamic system with time delay. In order simplify the system, it assumed that the system is non-dynamic system. When the collected data granularity is dense enough, MRNN can be used to model the chiller plants. MRNN model structure see Figure A.4 the model structure, we constrain part of the weight parameters to be non-negative (stU 2 0, V 0, W Z 0, D1 2 0, D2 2 0, D3 2 0) to ensure the monotonicity of input and output The performance and timing are monotonic, and a mask layer added to the input layer. Use the Ptrelu activation function, and the output layer is Relu. D1, D2, D3 are the weights of the pass through layer to improve the fitting ability the network.
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Figure A.4: MRNN.
13


Reviews:
[{"review_id": "xmdTuKuLj81", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 2, "confidence": 4, "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 presents how to utilize the domain knowledge which might be useful for solving problems in developing a deep neural network with the forms of variable constraints or objective functions. And the proposed method, called 'Monotonic Neural Network', was applied to the application of chiller plant optimization. Compared to the method (MLP), the proposed method showed better performance in the accuracy measure. \n\nThough this article shows how the machine learning model can be guided with prior knowledge in the real-world application, it is hard to say that this paper is above the accept line as there are lots of rooms for improvement.\n\nCompleteness\n- No conclusion: To make the research more consolidate, authors need to consider putting conclusions of their works including summary, reemphasizing what they found, some key points, or future works. By just finishing with 'Experiments' in this article, readers may wonder what are the main points of arguments from authors.\n- Explanations for abbreviation: As the paper deals with the application of  'Chiller plant', using technical terms is inevitable(e.g PUE, We-Bulb Temperature, DC, or CHWP). However, authors need to bring the meaning of the terms from the appendix into the main article so that readers can glimpse easily when the unknown terms are shown. \n\nApplicability\n- Its applicability is limited only to the same problem from the same facilities. Authors may consider making their proposed method more general so that practitioners from other industries who want to combine their domain knowledge into their problem can get ways or insights from this study.\n- No specific information about the proposed model structure found. How many layers(k) are used and why? What is the size of the layer weights? How are hyperparameters set from how many candidates? \n\nSuperiority\n- Authors may compare their works to other state-of-the-art approaches to present the superiority of them. Results showing better than MLP may not appeal to readers. \n\nTypo/correction\n- Authors need to check the correctness of sentences. Especially, the first paragraph of Section 2 has several misuses of capital letters. And the fifth line(Although Some... ) is not a complete sentence.\n- In section 3, Consider -> Considering\n- In section 2(Third paragraph), state of art -> state of the art\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "mC2GEEYcdnB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "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": "Summary:\n\nThis paper tackles the energy optimization problem of chiller plants, which are the main equipment of a cooling system. This paper aims to learn the non-linear mapping between the control parameters (of a cooling system) and energy consumption. The key idea of this paper is to incorporate domain knowledge into the training of neural networks. Specifically, monotonic constraints are proposed to constrain the learning of the non-linear mapping so that physical laws are satisfied. The authors propose two methods to implement monotonicity: hard-MNN and soft-MNN.\n\nHard-MNN treats monotonicity as hard constraints and uses a Monotonic Neural Network to learn the non-linear mapping. Soft-MNN treats monotonicity as soft constraints and adds a monotonic loss into the training objective to encourage the learned mapping to be monotonic.  Experiments show that (1) both the hard-MNN and soft-MM outperforms multilayer perceptron (MLP), and (2) hard-MNN yields lower energy consumption than MLP.\n\nPros:\n\n+ The idea of incorporating domain knowledge into training neural nets is reasonable.  (But this is not a new idea.   E.g.,  \nKG-GAN: Knowledge-Guided Generative Adversarial Networks.)\n\n+ Appendix A.1 does a great job of giving an introduction about cooling systems and chiller plants.\n\nCons:\n\n- Some claims are unjustified.\n\nSection 2 states that “we make a theoretical analysis and …” However, I cannot find such analysis. Does the theoretical analysis mean Equation 3, 4, and Table 1?\n\nSection 2 states that “Compared with the above state of art method, MNN reduces the dependence on the amount of data, provides a more accurate function space, facilitates subsequent optimization steps and improves optimization performance.” However, the current experimental results are not strong enough to support these four claims.\n\nSpecifically, no experiments empirically compare performance in terms of different amounts of training data. There is no attempt to quantify the so-called “a more accurate function space.” No results show information about the optimization applied to the learned mapping function.\n\n- The technical method is not clearly described.\n\nAppendix A.3 gives an overview of how the energy optimization problem is decoupled into two sub-problems: identification and optimization.\n\nI appreciate the overview but also think there is room for improvement.\n\nThis part could be important since it gives a concrete high-level view of the problem formulation. In particular, this paper concentrates on the first sub-problem. However, it is not evident to see this point neither from the introduction nor from the beginning of the method section, making it hard for me to understand the problem formulation clearly in my first-pass reading.\n\nIn Section 4, three type models are introduced. However, their names and notations are not linked together, which makes the reading very difficult. It would be better if the notations are shown together with their corresponding names like cooling/chilled water pump power model (P_{COWP} and P_{CHWP}), cooling tower power model(P_{CT}), and chiller power model (P_{CH}).\n\nThe connections between math equations are loose, which makes the technical method hard to understand. For example,\n\n(1) the connection between y in Equation 2 and y_{CH} in Equation 4 is not clear.\n(2) The connection between y_{CH} in Equation 4 and x_{CH} in Equation 5 is also not clear enough.\n(3) (c,f_{chiller}(x)) mentioned in Section 4.3 is somewhat isolated, its relationship with the other equations are not evident.\n\nThe last five lines of Section 4.3 (below Figure 4.4) are hard to understand. Overall, the technical writing of Section 4 could be further improved.\n\nFigure 4.1 and 4.4 lack a detailed explanation and thus are hard to understand. What do f_fan, cow_pump, P3, P1+P2+P3 in the figure mean, respectively? What is the difference between (a) and (b)? Why (a) is called identification hyperplane, while (b) is called optimization hyperplane?\n\nMost of the math notations are not explicitly explained in the paragraph. I had difficulty reading the mathematical equations until I found the table of notations in the Appendix (Table 2). However, Table 2 is not referenced throughout the paper.\n\n- Insufficient experiments in terms of setting, details, and comparisons.\n\nThe Experiments section lacks a detailed description of dataset information, training settings, implementation details. The current experiments look impossible to reproduce the results. What is the general introduction of the dataset used in the experiments? Do you split the data into training, validation, and testing set? What is the number of data examples of each subset? How to train the proposed model? What is the detailed network architecture of the MLP and the proposed MNN?\n\nIn the first experiment (Figure 5.1), what is MAPE, and how to calculate it? What is DC mentioned in Figure 5.1?\nThe second experiment (Figure 5.2) is unclear in its purpose, comparison method, metric, and effectiveness.\nIt is hard to judge the effectiveness of the proposed method based on the experiments.\nWhy do we need to consider experimenting with different wet-bulb temperatures?\n\nWhat is the key difference in terms of the experiment setting compared with the previous experiment? The 1st experiment focuses on internal control, while the 2nd experiment focuses on external control?\nCan the authors elaborate more on these two experiments (Figure 5.1 and 5.2)?\nWhat is PUE, and how to calculate it?\nWhat do the dots and lines in Figure 5.2 mean, respectively?\nWhat is MLP with local PID? PID is not explained throughout the paper.\nIs improving the average PUE by 1.5% significant?\nWhy no results of soft-MNN in this experiment?\n\nAs the related work mentioned, “linear regression is the mostly used modeling method in real-world …,” a baseline comparison with linear regression looks essential to let readers know the baseline performance. However, the current manuscript compares with MLP only. Is there any reason that prevents from comparing with linear regression?\n\nThis paper proposes a new activation function (Equation 11) and two extensions of the proposed monotonic neural networks (MNN): partial-MNN and recurrent-MNN. However, no results are shown to demonstrate their effectiveness.", "extra_scores": {}, "extra_text": {}}, {"review_id": "c_LBEScukzj", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "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": "The paper uses monotonic neural networks to estimate the energy consumption of chilling plants. This can be used to save energy by using the neural network in Model-Based Optimization.\n\nOn the positive side, the paper actually tests a monotonic neural network in MBO on a real chilling plant, dropping the PUE from 2% (from 1.54 to 1.52, e.g.). \n\nOn the negative side, all of the fundamental ideas in the submission are quite old:\n*  The idea of penalizing non-monotonic behavior in neural networks was first published in [1].\n*  The idea of forcing a monotonic structure on a neural network was first published in [2]. The submission references [3], which enforces convexity (not monotonicity).\n*  The idea of using a neural network to predict the energy consumption of a chilling plant was published in [4], although may predate that.\n\nNow the submission does not exactly reproduce these older ideas: e.g., the monotonicity penalty in [1] is quadratic, while in the submission it's either linear, or a log loss mapped through a sigmoid. However, no comparison between these ideas was tested, so it is unclear whether the difference from the old work is significant.\n\nThis paper would be of interest to a small segment of engineers who are interested in optimizing chilling plants. It could be made of wider interest if either (a) some stronger mathematical statements could be made, or (b) if more than one empirical experiment were carried out, perhaps across different types of systems. Perhaps the authors could try a system identification benchmark, e.g., [5]?\n\nREFERENCES\n1. Sill, Joseph, and Yaser S. Abu-Mostafa. \"Monotonicity hints.\" In Advances in neural information processing systems, pp. 634-640. 1997.\n2. Sill, Joseph. \"Monotonic networks.\" In Advances in neural information processing systems, pp. 661-667. 1998.\n3. Amos, Brandon, Lei Xu, and J. Zico Kolter. \"Input convex neural networks.\" In International Conference on Machine Learning, pp. 146-155. 2017.\n4. Gao, Jim. \"Machine learning applications for data center optimization.\" (2014).\n5. nonlinearbenchmark.org ", "extra_scores": {}, "extra_text": {}}, {"review_id": "9bwIRzA5h2u", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "Many thanks to the authors for their submission which aims at combining data-driven approaches with model-based ones for optimising centralised cooling systems (chiller plants). The overarching aim of this paper has been well studied before; given one aims at incorporating physical constraints, one needs a solid understanding of the underlying physical system, hence how one adds such a constraint to the model requires deep investigations. The machine learning part of this paper is also not something new. To match the physical constraints of the problem in hand authors incorporate monotonicity to the Neural Network model. Other than that, the paper does not offer a comprehensive explanation on what the novelty and actual contributions are, nor does it provide a solid experimental set-up. Please consider some of my comments below:\n\nMajor comments:\n--No conclusion or discussion has been provided. That makes the paper incomplete\n--Experimental section is rather inadequate and rushed up; the results are not explained properly; no concrete information on the dataset used has been provided to contextualise the experimental design.\n--Paper is heavily skewed towards providing too much background information and rather simplistic information on incorporating physical constraints on to the loss function of a neural-net.\n\nMinor comments: Quite a few punctuation errors exist. Some sentences are not finished or finish with a full stop where it shouldn't. Please consider proof-reading the manuscript carefully for correcting such typos.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Thank you for your submission to ICLR.  The reviewers unanimously felt that there were substantial issues with this work, owing to the fact that both the techniques and applications have been considered in a great deal of previous work.  Furthermore, the manuscript itself needs substantial amounts of revision before being suitable for publication.  As there was no response to these points during the rebuttal period, it seems clear that the paper can't be accepted in its current form.
Decision: Reject