online physics problem solving

v = 0 m/s

v = 88.3 m/s

( 112 m/s) 2 = (0 m/s) 2 + 2*(a)*(398 m)

12544 m 2 /s 2 = 0 m 2 /s 2 + (796 m)*a

12544 m 2 /s 2 = (796 m)*a

(12544 m 2 /s 2 )/(796 m) = a

a = 15.8 m/s 2

Return to Problem 19

v f 2 = v i 2 + 2*a*d

(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(91.5 m)

0 m 2 /s 2 = v i 2 - 1793 m 2 /s 2

1793 m 2 /s 2 = v i 2

v i = 42.3 m/s

Now convert from m/s to mi/hr:

v i = 42.3 m/s * (2.23 mi/hr)/(1 m/s)

v i = 94.4 mi/hr

Return to Problem 20

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Discover Frequently Asked Math Questions and Their Answers

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Calculating faster: Coupling AI with fundamental physics

by Stefanie Terp, Technical University of Berlin

Atoms are complex quantum systems consisting of a positively charged nucleus surrounded by negatively charged electrons. When multiple atoms come together to form a molecule, the electrons of the constituent atoms interact in a complicated manner, making the computer simulation of molecules one of the hardest problems in modern science.

Researchers from the Berlin Institute for the Foundations of Learning and Data (BIFOLD) at TU Berlin and Google DeepMind have now developed a novel machine learning algorithm which enables highly accurate simulations of the dynamics of a single or multiple molecule on long time-scales. Their work has now been published in Nature Communications .

These so-called molecular dynamics simulations are important to understand the properties of molecules and materials and have potential applications in drug development and material design (e.g. for use in solar panels and batteries). Traditional methods to compute the interactions of electrons rely on finding solutions of the so-called Schrödinger equation.

The Schrödinger equation describes the energy levels that a quantum system—e.g. atoms or molecules—can assume. This is a notoriously difficult task, and finding a solution for molecules containing more than a few dozen atoms may take several days—even on powerful computers. To make matters worse, for running molecular dynamics simulations over long time-scales, the Schrödinger equation needs to be solved thousands or even millions of times, making the computational cost quickly exceed the compute resources that are available today.

"The simulation of such interactions and the resulting predictions for complex processes like protein folding or the binding between individual molecules is a long-held dream of many chemists and material scientists, and would save many expensive and labor-intensive experiments," explains BIFOLD researcher Thorben Frank.

In recent years, machine learning (ML) methods have brought this dream within reach. Instead of explicitly solving the Schrödinger equation, they can learn to directly predict the overall outcome of the relevant electronic interactions at the atomistic level, with greatly reduced computational cost.

The difficulty is then shifted to finding efficient algorithms for "teaching" the machine learning system how the electrons interact without modeling them explicitly. To reduce the complexity of this task, many learning algorithms use the fact that physical systems follow so-called invariances.

Simply put, certain properties of molecules stay the same when molecules are moved in space but the relative distances between individual atoms stay fixed—meaning the machine does not need to learn anything new in these cases. However, the way these invariances are typically incorporated into ML models is computationally expensive, ultimately limiting the speed with which the models can perform molecular dynamics simulations.

To address this shortcoming, the BIFOLD scientists have devised a new learning algorithm that decouples invariances from other information about a chemical system at the outset. Unlike previous methods that required extracting invariant components from each operation within the model, this new approach simplifies the process. Now, the ML model can reserve the most complex operations for the physical information that really matters and drastically reduce the overall computational cost.

"Simulations that required months or even years of computation on high-performance computer clusters, can now be performed within a few days on a single computer node. The leap in efficiency allows long-time scale simulations, which are necessary for understanding the structure, dynamics and functioning of atomistic systems. It thus enables deeper insights into the most complex and fundamental processes of nature," says BIFOLD researcher Dr. Stefan Chmiela who spearheaded the research project.

In the future, the accurate simulation of the interaction of molecules with proteins in the human body could allow researchers to develop new drugs without the need to perform experiments—saving time and money while at the same time being more environmentally friendly.

To showcase potential applications of the algorithm, the team used the new ML method to identify the most stable version of docosahexaenoic acid, a fatty acid which is a primary structural component in the human brain. This task requires scanning tens of thousands of potential candidates with high accuracy. So far, such an analysis would have been infeasible with traditional quantum mechanical methods.

As noted by Prof. Dr. Klaus-Robert Müller, BIFOLD co-director and Principal Scientist at Google DeepMind, "This work demonstrates the potential of combining advanced machine learning techniques with physical principles to overcome long standing challenges in computational chemistry. It continues a critical line of research which puts focus on scaling ML approaches towards realistic chemical systems of practical interest."

Dr. Oliver Unke, Senior Research scientist at Google DeepMind comments, "Earlier this year, we succeeded in scaling models to thousands of atoms, but with new advancements like this, moving to even larger numbers of atoms may become possible."

While simulations with tens to hundreds of thousands of atoms are now becoming accessible, some structures consist of millions of atoms or more. The next generation of algorithms will need to be able to simulate such system sizes accurately, which requires a correct description of additional, complex, long-range physical interactions.

Journal information: Nature Communications

Provided by Technical University of Berlin

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Introduction

The Online Physics Olympiad (OPhO) is one of the largest online run physics olympiads in the world. This contest is a team based contest with each team consisting of 1-3 high school students. Undergraduate students in university may also participate, but they will be in a separate division and are not able to qualify for prizes.

Visit the official website at: https://opho.physoly.tech/

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Physical Review Accelerators and Beams

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Bayesian optimization algorithms for accelerator physics

Ryan roussel et al., phys. rev. accel. beams 27 , 084801 – published 6 august 2024.

• No Citing Articles
• INTRODUCTION
• BACKGROUND AND MOTIVATION
• GAUSSIAN PROCESS MODELING
• ACQUISITION FUNCTION DEFINITION
• ACQUISITION FUNCTION OPTIMIZATION
• ACKNOWLEDGMENTS

Accelerator physics relies on numerical algorithms to solve optimization problems in online accelerator control and tasks such as experimental design and model calibration in simulations. The effectiveness of optimization algorithms in discovering ideal solutions for complex challenges with limited resources often determines the problem complexity these methods can address. The accelerator physics community has recognized the advantages of Bayesian optimization algorithms, which leverage statistical surrogate models of objective functions to effectively address complex optimization challenges, especially in the presence of noise during accelerator operation and in resource-intensive physics simulations. In this review article, we offer a conceptual overview of applying Bayesian optimization techniques toward solving optimization problems in accelerator physics. We begin by providing a straightforward explanation of the essential components that make up Bayesian optimization techniques. We then give an overview of current and previous work applying and modifying these techniques to solve accelerator physics challenges. Finally, we explore practical implementation strategies for Bayesian optimization algorithms to maximize their performance, enabling users to effectively address complex optimization challenges in real-time beam control and accelerator design.

• Received 9 December 2023
• Accepted 3 June 2024

DOI: https://doi.org/10.1103/PhysRevAccelBeams.27.084801

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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Overview of challenges in using optimization algorithms for online accelerator control. Accelerator control algorithms make decisions about setting a wide variety of accelerator parameters in order to control beam parameters at target locations. Optimal decision making takes into account limited online accelerator measurements, as well as various sources of prior knowledge about the accelerator, including previous measurements, physics simulations, and physics principals. Optimization must also consider complicated aspects of realistic accelerator operation including external conditions, feedback systems, safety constraints, and repeatability errors.

Overview of optimization challenges in accelerator physics simulations. Ideal algorithms aim to minimize the computational cost of performing optimization by orchestrating parallel simulation evaluations at multiple fidelities ranging from analytical models to high-fidelity (computationally expensive) simulations. Correlations between simulation predictions at different fidelities can be leveraged to reduce the number of high-fidelity simulation evaluations needed to find an ideal solution at the highest-fidelity level.

Illustration of the Bayesian optimization process to find the maximum of a simple function. A Gaussian process (GP) model makes predictions of the function value (solid blue line) along with associated uncertainties (blue shading) based on previously collected data. An acquisition function then uses the GP model to predict the “value” of making potential future measurements, balancing both exploration and exploitation. The next observation is chosen by maximizing the acquisition function in parameter space. This process is repeated iteratively until optimization goals have been reached.

Illustration of Bayesian regression using a linear model f ( x ) = w 1 x + w 0 . (a,d,g) Posterior probability density of the linear weights { w 1 , w 0 } conditioned on N observations of the function y = f ( x ) + ε . (b,e,h) Model predictions using random samples of { w 1 , w 0 } drawn from the posterior probability distribution. (c,f,i) Predictive mean (solid line) and 90% uncertainty intervals (shading) of the posterior model. Red cross and black dashes denote true parameters and values of the function f ( x ) , respectively. Reproduced with permission from [ 39 ].

Illustration of GP model predictions. (a) Prior model prediction of the function mean (solid blue line) and confidence interval (blue shading) at a set of test points in parameter space. The probability of the output value y at any given test point x * is a normal distribution. (b) The posterior GP model also predicts normal probability distributions at each test point, conditioned on the dataset D . (c) Individual function samples can also be drawn from the posterior GP model and can be used for Monte Carlo computations of function quantities.

Visualization of how the length scale hyperparameter l effects GP modeling. Three GP models are trained on the same dataset using a Matérn kernel with fixed length scales of (a) 0.1, (b) 1, and (c) 10. Remaining hyperparameters are trained by maximizing the marginal log-likelihood.

Examples of GP modeling with varying treatment of measurement noise. (a) Shows a GP model containing zero noise, forcing the GP prediction to fit experimental data exactly. (b) Shows a GP model trained on the same data with a fixed (homoskedastic) noise parameter. (c) Illustrates a GP model incorporating heteroskedastic noise, where the data variance for each point is explicitly specified.

Illustration of the improvement in prediction accuracy that can be gained by including expected correlations, such as those that arise from adjacent quadrupoles, into the GP kernel design. Here, a 2D function has input correlations that are similar to what one might observe between adjacent quadrupoles (a). For a fixed set of training data points (shown in orange), a GP model using an uncorrelated kernel (b) produces less accurate posterior predictions of the true function than a model with an accurate correlated kernel (c). In the context of BO, learning a more accurate model with fewer data training points translates to faster convergence in optimization.

Illustration of nonzero prior mean. In the absence of local data, the mean of the posterior distributions reverts to (a) zero or (b) the nonzero prior mean. The variance remains unchanged.

Transmission optimization at ATLAS subsection using different prior mean functions. Solid and dashed lines depict the medians and the shaded areas depict the corresponding 90% confidence levels across 10 to 20 runs. Reproduced from [ 24 ].

Example of using log transformations in GP modeling for strictly positive output values. Data in real space (a) are transformed to log space before fitting a GP model (b). Samples drawn from the GP model in log space can then be transformed back into real space to make GP predictions. The resulting likelihood in real space is then a log-normal distribution which is strictly positive.

Simulated application of standard and time-aware BO in a drifting trajectory stabilization problem. Simple BO settles on the mean value of the oscillations. ABO-ISO (isotropic) follows the changes but lags them because it only uses isotropic (local) kernel. ABO-SM (spectral mixture) captures long-range correlations and eventually correctly predicts necessary future changes in phase. By default, ABO-SM continues to explore around maximum value for optimization, producing a small step jitter. It can be eliminated by using the posterior mean as the acquisition function at the cost of convergence speed.

Illustration of the prediction of a multifidelity Gaussian process, by comparing (a) a single-fidelity Gaussian process trained only on high-fidelity data, and (b),(c) a multifidelity Gaussian process trained on both high-fidelity and low-fidelity data, in the case where (b) high-fidelity and low-fidelity data are highly correlated, as well as (c) high-fidelity data and low-fidelity data are largely uncorrelated. In this particular example, the multifidelity GP is a multitask GP [ 61 ], as implemented in the library b o t orch. Dashed lines denote ground truth values of the low- and high-fidelity functions.

Demonstration of combining GP models with a differentiable physics model of magnetic hysteresis. (a) Measured beam charge after passing through an aperture in the APS injector is plotted over three cycles of varying the current in an upstream quadrupole. Transmitted beam charge measurements are not repeatable due to hysteresis effects in the upstream quadrupole. (b) GP modeling with differentiable hysteresis model included accurately predicts beam charge over multiple hysteresis cycles with improved (reduced) uncertainty predictions. Reproduced from [ 28 ].

Examples of the EI and UCB acquisition functions for objective function maximization given the same GP model and training data. (a) EI acquisition function, where the dashed horizontal line denotes the best previously observed value f ( x * ) . (b) UCB acquisition function.

Example of sampling behavior of Bayesian exploration (BE). (a) The BE acquisition function is maximized at locations in parameter space where the model uncertainty is highest, usually at locations farthest away from previous measurements. (b) In cases where the function is less sensitive to one parameter ( x 2 in this example), the model uncertainty is smaller along that axis, resulting in less frequent sampling along that dimension.

Comparison between different constrained Bayesian optimization algorithms. (a) Weighting the acquisition function by the probability of satisfying the constraining function [ 73 ]. (b) Acquisition function optimization within a safe set using MoSaOpt in exploitation mode [ 32 ] and (c) SafeOpt [ 75 ]. (d) The constraint function, where valid regions satisfy c ( x ) > 0 .

Summary of multiobjective BO (MOBO) using expected hypervolume improvement (EHVI). (a) Given Pareto front P and corresponding hypervolume H , the increase in hypervolume H I due to a new measurement y is given by the shaded green area. (b) Comparison between multiobjective optimization algorithms for optimizing the AWA injector problem. NSGA-II is a standard evolutionary algorithm [ 81 ], I-NN is surrogate model assisted NSGA-II [ 54 ]. (c) Projected hypervolume after a set number of MOBO iterations with insets showing hypervolume improvement due to fill in points (i) and measurement of newly dominant points (ii). Reproduced from [ 34 ].

Visualization of the BAX process for beam steering through quadrupole magnets. (a) Experimental measurements are used to build a GP model of the horizontal beam centroid position at a downstream screen C x as a function of the quadrupole strength and steering parameter. Note that the GP model is built with a first-order polynomial kernel, constraining predictions to planar surfaces. Dashed lines denote cross sections of the GP model shown in (b). (c) The BAX acquisition function that predicts the information gained about the ideal steering current by making future measurements.

Demonstration of proximal biasing effects during Bayesian exploration (BE) of the constrained TNK test problem. (a) Normal BE. (b) BE using proximal biasing with l = 0.1 . The green arrow highlights a step where a larger jump in parameter space was allowed by proximal biasing. Reproduced from [ 86 ].

One-dimensional visualization of trust region BO (TuRBO) applied to a minimization problem with the UCB acquisition function. (a)–(d) Sequential evolution of the GP model and sampling pattern. Orange circles denote objective function measurements and green circles denote the most recent sequential measurement at each step.

Trust region BO (TuRBO), simplex, and UCB applied to the minimization of total losses (maximization of lifetime) at the ESRF-EBS storage ring. Adapted from [ 96 ].

Comparison of optimization performance between a local optimization algorithm (Nelder-Mead simplex), BO using the UCB acquisition function ( β = 2 ), and BO using the UCB acquisition strongly weighted toward exploration ( β = 100 ). All algorithms are initialized with a single observation at x = 0.75 and aim to minimize the objective function. (a)–(c) Observations of the objective function in parameter space for each algorithm. The dashed line denotes the true objective function. (d)–(f) Objective function values as a function of algorithm iteration. Note that simplex terminates after reaching a convergence criteria.

Comparison between GP modeling of hard and soft constraining functions. (a) GP modeling of a heaviside constraining function does not accurately predict constraint values due to a single sharp feature that cannot be learned without dense sampling on either side of the constraint boundary. (b) Smooth constraining functions with a single characteristic length scale are more accurately modeled with GP modeling. Inset: Visualization of bounding box constraint function f ( x ) = max i { | | C − S i ( x ) | | } used to keep beam distributions inside an ROI, where r is the radius of a circular ROI, C is the center coordinates of the ROI, and S i are corner coordinates of a bounding box around the beam.

Comparison between GP modeling of the two-dimensional sphere function f ( x 1 , x 2 ) = x 1 2 + x 2 2 with and without interpolated measurements. (a) Shows the posterior mean of the GP model with four measurements taken sequentially. (b) Shows the same four measurements taken sequentially but with interpolated points in between each measurement. Incorporating interpolated points in the dataset leads to higher modeling accuracy, leading to accurate identification of the sphere function minimum at the origin.

Performance scaling with dataset size for b o t orch/ gp y t orch ( 0.9.4 / 1.11 ) libraries on a single-objective optimization run. Synthetic five-variable quadratic objective was used with Monte Carlo version of UCB acquisition function and 100 Adam optimizer iterations. GPU memory usage is only applicable to GPU runs.

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• 07 August 2024

Physics solves a training problem for artificial neural networks

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Huge changes are already under way in health care, industry and education as a result of advances in artificial intelligence (AI), but the costs could outweigh the benefits if AI’s enormous energy consumption is not reined in. The problem is that AI relies heavily on deep neural networks, which are layered algorithms that involve millions or even billions of computations, requiring massive, energy-hungry access to the memories in conventional computers. One possible solution is to replace the computers with systems that more closely reflect the physical structure of biological neural networks, but such systems are typically incapable of performing one of the main steps in training the network. In a paper in Nature , Xue et al . 1 report an ingenious workaround that uses physics to overcome the problem.

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Scellier, B., Ernoult, M., Kendall, J. & Kumar, S. Adv. Neural Inf. Proc. Syst . 36 , 52705–52731 (2023); available at https://go.nature.com/4fmijnb .

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