Optimizing High-Dimensional Physics Simulations
via Composite Bayesian Optimization

Wesley J. Maddox∗
New York University
wjm363@nyu.edu

Qing Feng
Facebook
qingfeng@fb.com

Max Balandat
Facebookbalandat@fb.com

Abstract
Physical simulation-based optimization is a common task in science and engineering. Many such simulations produce image or tensor based outputs where thedesired objective is a function of that image with respect to a high-dimensional parameter space. We develop a Bayesian optimization method leveraging tensor-based
Gaussian process surrogates and trust region Bayesian optimization to effectively
model the image outputs and to efficiently optimize these types of simulations,
including an optical design problem and a radio-frequency tower configuration
problem.

1

Introduction

Many design problems in the physical sciences require running simulations to evaluate a new design.
Examples abound in material science (Zhang et al., 2020), fluid dynamics (Anderson and Wendt,
1995), and optics (O’Shea et al., 2004). Typically, these simulations generate high-dimensional
outputs, often in the form of image- or other tensor-structured formats. This usually requires
substantial computational effort and simulations may take a long time to run. Optimizing designs
thus presents a formidable challenge, and using sample-efficient optimization methods is crucial.
In this work, we solve a challenging design problem, namely optimizing the geometry and gratingsof a diffractive optical element. In our example the design is parameterized by 177 parameters; eachsimulation takes about one hour to run and produces an x × 16 × 16 output representing the projected
image. Our goal is to explore the efficient frontier between efficiency (how much light is delivered)and uniformity (how uniform is the output) of the image generated by the device.
One approach to optimizing such a problem is to build a surrogate model of the simulation that ischeap/quick to evaluate, and perform the optimization based on this surrogate. Bayesian Optimization(BO) is an established method following this paradigm, and has been successfully applied in wide
range of settings, including many in the physical sciences (Packwood, 2017; Zhang et al., 2020; Duriset al., 2020). Historically, BO has been restricted to relatively low-dimensional design spaces, a small
number of evaluations, and a single scalar outcome. Recently, Maddox et al. (2021a) developed ascalable approach to composite BO with high-dimensional outputs that relies on an efficient sampling
scheme for High-Order Gaussian Process (HOGP) models that provide a probabilistic model overtensor-structured outputs. However, their method suffers from the same scaling challenges in termsof the dimension of the design space as standard GP models do (Eriksson et al., 2019; Eriksson and
Poloczek, 2021), and is thus not applicable to our 177-dimensional optics optimization problem.
In this work, we combine the model and sampling scheme from Maddox et al. (2021a) with the recent
MORBO algorithm for high-dimensional multi-objective Bayesian Optimization from Daulton et al.
(2021). We overcome additional memory scalability challenges by employing a mixed-precision compute paradigm and batching computations, enabling improved performance over existing baselines.
∗

Work performed during an internship at Facebook.

Fourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

2

Methodology

2.1

High-Order Gaussian Process for Image Pixel Prediction

To perform composite Bayesian Optimization in the space of large images, we use the high-order
Gaussian Process (HOGP) proposed by Zhe et al. (2019) to model the images from the optics
simulator given a design configuration. This model extends the traditional multi-task Gaussian
processes (MTGPs) and can more efficiently handle high-dimensional correlated outputs.
The HOGP model tensorizes the image outputs y ∈ Rn×d1 ×···×dk , and learns latent features of eachtensor element to capture their correlations. It assumes the covariance between any two outputs, y, y0 ,is the given as the elementwise product of the output indices. The covariance function isk([x, i1 , · · · , ik ], [x0 , j1 , · · · , jk ]) = k(x, x0 )k(v1 , v10 ) · · · k(vk , vk0 ),where i1 , · · · , ik are the indices for the output tensor, v1 , · · · , vk are the latent parameters, andk(x, x0 ) is the kernel over the parameter space. Thus, the task covariance function in the MTGP
framework is represented as a chain of Kronecker products so that the GP prior is vec(y) ∼N (0, KXX ⊗ K2 ⊗ · · · ⊗ Kk ). See Appendix B.1 for more discussions.
2.2

Composite Multi-objective Optimization over High-dimensional Search Space

Given the predicted images from HOGP, we can perform composite BO (Astudillo and Frazier, 2019)that optimizes composite objectives of the form maxx g(h(x)), where h : Rd → Rd1 ×···×dk is theexpensive simulation that produces a tensor as output, and g : Rd1 ×···×dk → Ro is the deterministic
function to compute goal metric e.g. efficiency of image outputs.
The optical design problem poses two primary challenges: 1) the design space is high-dimensionalwith 177 parameters to optimize; 2) the goal is to find the set of optimal tradeoffs between the twocompeting objectives (efficiency and uniformity) rather than optimizing a single objective. GP-based
BO usually works well for problems with search spaces having less than 20 or so parameters; but
does not scale well to high-dimensional parameter spaces: as distances grow larger and model
uncertainty towards the boundary of the search space increases, BO tends to over-explore. To avoid
this issue, Eriksson et al. (2019) introduced trust region Bayesian optimization (TRBO) that performs
optimization in smaller trust regions that evolve across the search space. More recently, (MORBO)
Daulton et al. (2021) extended this work to the multi-objective setting. While MORBO handles
high-dimensional parameter spaces well, Daulton et al. (2021) only considered problems with few
outcomes and non-composite settings. In this work we employ MORBO in conjunction with the
improved HOGP model to perform composite BO over high-dimensional outcome spaces (images).
2.3

Efficient Posterior Sampling for MORBO with HOGP

MORBO/TRBO construct trust regions and perform local modeling and optimization. Thus, we build
HOGP using observations inside each trust region. Since HOGP is a sample-efficient model, we
can achieve good predictions of image pixels and the aggregated goal metrics (see Figure 4a for an
example), and also reduce computational cost of using all the data points.
Both MORBO and TRBO rely on Thompson sampling for optimizing acquisition functions, which
is implemented as drawing a large numbers of GP posterior samples evaluated at many discrete
candidates, xtest . Drawing posterior samples from HOGPs can be computationally expensive andeven be intractable for high-dimensional outputs such as images. The time complexity to sample
over all outputs (tasks) and all new data points is multiplicative in the number of outputs O((n3 +Qd
n3test ) i=1 d3i ) (Maddox et al., 2021a). What’s more, storing k posterior samples at ntest test points
requires storage of k × ntest × n × d1 × · · · × dk tensors, which quickly becomes problematic on amemory-restricted GPU. To make it feasible to combine HOGP with MORBO/TRBO, we leverage
the efficient posterior sampling developed by Maddox et al. (2021a) to reduce the time complexity
Pdto O((n3 + n3test ) + i=1 d3i ) and further propose two remedies to improve memory complexity of
posterior sampling. See Appendix B.2 for more technical details of the posterior sampling proposedby Maddox et al. (2021a).
Improving memory efficiency with batch and mix-precision computation First, we segment
test points into small batches and loop batches of n0  ntest test points, as they affect the size of the
2

20

40

60

80

Function Evaluations

100(a) Environmental,
t = 3 × 4, d = 4, q = 3.

1865

1
0
1
2
3
4

Log10(Regret)Log10(Regret)Log10(Regret)

RandomqEIHOGP + qEIHOGP + TRBO
GP + TRBO

0
1
2
3
4
5

1860
1855
1850

20

40

60

80

Function Evaluations

100(b) Environmental,
t = 5 × 10, d = 4, q = 3.

50

100

Function Evaluations

150(c) Cell-tower coverage,t = 2 × 50 × 50, d = 30, q = 1.

Figure 1: Benchmark traces (average across 20 runs with 95% confidence interval) for single objective
problems. HOGP+TRBO achieves the best performance on the environmental problems and performs
similarly as TRBO, obtaining the best reward on the cell-tower coverage problem.

Kronecker matrix vector multiplications more than the posterior samples, k. This enables smaller
matrix vector products and thus reduced memory overheads.
Second, we employ a mixed-precision computing paradigm and use half precision arithmetic tocompute the Kronecker matrix vector products when evaluating the posterior samples, while using
double precision for the numerically demanding matrix root computation of the potentially poorly
conditioned data covariance. Unlike previous work such as Gardner et al. (2018); Maddox et al.
(2021b) finding implementation difficulties when moving to lower precision arithmetic, we perform
all training in float and double precision arithmetic, and only compute the posterior Kronecker matrix
vector products (which are the memory intensive ones) in half precision. This enables accuratecomputations when necessary, while preserving much of the speedups that are theoretically gained byusing lower precision arithmetic (Micikevicius et al., 2017).

3

Results and Discussion

3.1

Single Objective Experiments

We evaluate our method (HOGP+TRBO) that performs composite TRBO with HOGP model on
three single objective optimization problems, and compare with four methods: quasi-random search(Random), expected improvement on the metric (qEI), current TRBO (GP+TRBO) and composite BO
with HOGP and expected improvement (HOGP+EI). All the results are the mean and 95% confidence
intervals across 20 trials. See Appendix C.1 for more experimental details.
Environmental Problem We evaluate on a spatial problem in which environmental pollutant
concentrations are observed on a 3 × 4 grid originally defined in Bliznyuk et al. (2008); Astudillo
and Frazier (2019) and an expanded 5 × 10 grid. The goal is to optimize a set of four parameters toachieve the true observed value by minimizing the mean squared error of the output grid to the output
grid of the true parameters. As shown in Figure 1a and Figure 1b, current TRBO achieves lower regret
compared with Random and qEI and our method further outperforms TRBO. This demonstrates theefficiency of performing composite Bayesian optimization with HOGP.
Cell-Tower Coverage Problem Following Maddox et al. (2021a); Dreifuerst et al. (2020), we
optimize the simulated “coverage map" resulting from the transmission power and down-tilt settingsof 15 cell towers (for a total of 30 parameters) based on a scalarized quality metric combining
signal power and inference at each location so as to maximize total coverage, while minimizing
total interference. To reduce model complexity, we down-sample the simulator output to 50 × 50,
initializing the optimization with 20 points. Figure 1c shows that our method and current TRBO
achieve the best performance.
3

(a) Example image outputs from simulations.

(b) Example optimized image outputs from simulations.

Figure 2: Example output images (un-optimized (a) and optimized (b) using our approach(HOGP+MORBO) on the diffractive optical element problem; pixel values are displayed on the same
scale per panel. The optimized channels are significantly brighter and are smoother at same time as
would be expected from optimizing both the efficiency and uniformity metrics.

0.1050GP
HOGP

0.1025Hypervolume

15.0
12.5
10.0
7.5
5.0
2.5
0.0

Uniformity

0.10000.097500.0950

HOGP + MORBO
GP + MORBO
1000 2000 3000 4000 5000

0.09250.046

Function Evaluations

0.048

0.050

Efficiency

0.052

(b)

(a)

Figure 3: Performance on the multi-objective optical design problem. (a) HOGP+MORBO outperforms GP+MORBO in the early iterations and achieves similar maximum hypervolume at the end.
(b) Pareto frontiers (metrics are standardized so that lower is better for both metrics) over 20 trials.
HOGP+MORB can push more on optimizing uniformity which is harder to model.

3.2

Multi-Objective Design of a Diffractive Optical Element

We compare our multi-objective optimization approach (HOGP+MORBO) with current MORBO(GP+MORBO) (Daulton et al., 2021) on the 177-dimensional optics optimization problem. The
simulations generating the outputs of the elements are computed using a custom physics simulationengine. In order to evaluate the optimization performance at reasonable computational cost, we fit
a neural network surrogate model from the input parameterization to the image output based on alarge number of simulation runs sampled from the design space. In the benchmarks, we evaluate
designs based on this neural network surrogate rather than the real physics simulation engine. See
Appendix C.2 for the details of optical experimental setup. The goal is to jointly minimize twogoal metrics, efficiency and uniformity, used for measuring displayed image quality. We assess theperformance based on the maximum achieved hyper-volume.
Figure 2b demonstrates the substantial improvements on the displayed images through optimizing
with HOGP+MORBO. The optimized images are much brighter and are also smoother compared tothe un-optimized outputs. Figure 3a shows that our approach tends to outperform current MORBO
in the earlier stage of the optimization and reaches to a similar performance at the end. We further
visualize the Pareto frontiers (metrics are standardized) across 20 runs in Figure 3b. It can be seen thatthe HOGP can explore along the uniformity metric more efficiently than the GP model, but explores
less on the efficiency metric. Since the uniformity metric (defined as a ratio between extreme pixel
values) is a harder to model than efficiency (which is closely related to the average pixel brightness),
this again suggests the sample efficiency of HOGP from modeling the image outputs directly andlearning the latent correlation structure across image pixels to transfer information.
4

Acknowledgements
WJM was partially supported by a NSF Graduate Research Fellowship under NSF IIS-1951856.
We’d like to thank Dominic Meiser, Ningfeng Huang, Eytan Bakshy, and David Eriksson.

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Checklist
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes](b) Did you describe the limitations of your work? [Yes](c) Did you discuss any potential negative societal impacts of your work? [Yes] None
expected.
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(a) Did you state the full set of assumptions of all theoretical results? [N/A](b) Did you include complete proofs of all theoretical results? [N/A]
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(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] .
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [Yes](c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Error bars are two standard deviations of the mean across
20 random seeds.
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [Yes] We used a single 16GB Nvidia
GPU for each experiment.
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(a) If your work uses existing assets, did you cite the creators? [Yes](b) Did you mention the license of the assets? [Yes] Single objective problems are MIT
License; see further description in Appendix of Maddox et al. (2021a). Optical design
problem is currently proprietary.
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(d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [N/A](e) Did you discuss whether the data you are using/curating contains personally identifiable
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6

ABroader Impact Statement

The method introduced in this paper provides a sample-efficient solution to the challenging design
problems in the physical sciences. By overcoming the scalability blockers in leveraging tensor-based
Gaussian Process model and trust region Bayesian Optimization, we unlock the possibilities ofconducting optimizations in high-dimensional parameter spaces and output spaces. We do not expect
any negative social impacts from our work.

B

Methodological Details

B.1

Kronecker Matrix Vector Products

The key aspect of efficiency in the HOGP comes from Kronecker matrix vector products. Suchstructure allows it to capture complex output correlations and scale to high-dimensional outputs withno sparse approximation. Besides, kronecker matrix vector multiplies (MVMs) can be efficiencly
computed from:z = (K1 ⊗ K2 )vec(A) = vec(K2 AK1> );n1 ×n1
if K1 ∈ R
and K2 ∈ Rn2 ×n2 . As a result, computing z costs O(n21 + n22 + n1 n2 (n1 + n2 ))time. Note that we can recursively compute this structure across several matrix vector products.
Implementation wise, this involves reshaping the vector vecA to be a matrix and then computing amatrix matrix product, for example K2 A.
B.2

Efficient Posterior Sampling with the HOGP

We utilize the efficient posterior sampling mechanism for the HOGP model Maddox et al. (2021a)
proposes an efficient posterior sampling mechanism for general MTGPs and extends it to HOGPs
as a special case. The time complexity of this method is additive in the combination of tasks anddata points, rather than multiplicative, which allows us to perform time-efficient composite Bayesian
Optimization on the image outputs.
This method uses Matheron’s rule for sampling conditional Gaussian distributions (Goovaerts et al.,1997). For HOGP, f (xtest )|Y = y generated by Matheron’s rule can be represented as−1f¯ = f + Kxtest X ⊗di=2 Ki (KXX ⊗di=2 Ki ) + σ 2 I(y − Y − ),(1)where f ∼ N (0, K(xtest ,X),(xtest ,X) ⊗di=2 Ki ), that is drawn from the joint prior distribution that all
Pkkernel matrices are Kronecker structured, and  ∼ N (0, σ 2 I). Although the size of f¯ is still i=1 diand naively decomposing the posterior covariance matrix of the GP to produce posterior samples
Pk
Pk
Pk
would cost O(( i=1 di )3 ) time, the Matheron’s rule approach instead costs O( i=1 d3i + i=1 di )time to draw a single sample.

C

Experimental Details

C.1

Single Objective Experimental Details

In the benchmarks, the HOGP model and TRBO used reference implementations from their authors
with default settings. For qEI and GP+TRBO, we used a standard ARD Matern 5/2 kernel (Rasmussen and Williams, 2008) to model the aggregated goal metrics. For the methods using expected
improvement, we used the qExpectedImprovement acquisition implemented in BoTorch (Balandat
et al., 2020).
For the environmental problem, we followed the implementations of Balandat et al. (2020), Astudillo
and Frazier (2019), and used 8 random restarts, 256 MC samples, and 512 base samples, a batch limit
of 4, and an initialization batch limit of 8.
For the radio frequency coverage problem, we followed the evaluation setup in Maddox et al. (2021a).
We initialized with 20 points, down-sampled the two 241 × 241 outputs to 50 × 50 for simplicity, ran
the experiments over 20 random seeds and for 150 steps. We used 32 MC samples, 64 raw samples
with a batch limit of 4 and an initialization batch limit of 16.
7

C.2

Diffractive Optical Element Design

The surrogate model is a densenet style architecture and maps the 177 dimensional input parameters
into the 11 output images, each of which is of size 16 × 16. Example images that are optimized
are shown in Figure 2a. The goal of these design efforts is to jointly optimize the efficiency anduniformity of the images. Efficiency is computed as a weighted mean across an images, while
uniformity is computed as a ratio of the 99% percentile pixel in the image to the 1% percentile pixelin the image. To combine the metrics across images, we consider the log sum exp of all of the images
as a form of soft maximum.2 As the real simulation itself is noisy, we use a scale Binomial noise term
to inject noise into the image after being outputted from the surrogate NN mmodel: each image pixel
is drawn from a distribution following y ∼ Binomial(N, p ∗ 100)/(100 ∗ N ), where N = 5000. This
type of noise model matches the underlying physical structure of the simulator.
Experimental Details We used the MORBO implementation as referenced in Daulton et al. (2021).
Each trial was initialized with 400 quasi-random (Sobol) points and optimized the hypervolume
improvement with batch sizes of 50 over 5000 function evaluations. The results shown in Figure 3aare the mean and 95% confidence intervals (2 standard errors) of the achieved maximum hyper-volume
across 20 trials.

Efficiency

0.06001000x MSE: 0.01

0.05750.01000.05500.00750.0525

Uniformity
100x MSE: 0.76

Predicted

Predicted

0.01250.00500.05000.00250.04750.00000.04500.00250.04250.04000.0400 0.0425 0.0450 0.0475 0.0500 0.0525 0.0550 0.0575 0.0600

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175

Truth

Truth

(a) HOGP model fits on metrics after 5000 trials.

Efficiency
0.01750.200

1000x MSE: 0.0001

0.175

0.01500.150

0.01250.0100Predicted

Predicted

Uniformity

100x MSE: 0.83

0.00750.0050

0.125
0.100
0.075
0.050

0.00250.025

0.00000.000

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175
Truth

0.045

0.050

0.055
Truth

0.060

0.065

(b) GP model fits on metrics after 5000 trials.

Figure 4: Global surrogate model fits on the optics problem at the end of an optimization run (test /train sets are the same for both models). The GP surrogate is better at modelling efficiency, but isworse at modelling uniformity than the HOGP. The uniformity metric requires information across all
pixels to be modelled accurately, which makes it better suited to be modelled using a surrogate thatpredicts every pixel.
Benchmark Result Analysis We evaluate the out-of-sample prediction accuracy of standard GP
and HOGP model shown in Figures 4a and 4b. The plots compare the prediction of two goal metrics
on a holdout set of data from an optimization run. We see that while both models are reasonably
accurate at predicting both metrics, the GP is more accurate at predicting the efficiency metric,
explaining why it was able to explore across the Pareto frontier better on that metric. By comparison,
2

Please note that all scales of metrics in plots are normalized to be approximately [0, 1].

8

the HOGP is able to use the information of all pixels to better predict the uniformity metric, which is
tougher to model and more bi-modal — we truncated the test set to remove the outliers.

9