Veronica Saz Ulibarrena

Thesis Chapters

Ph.D. Thesis Chapters and Bibliography:

Individual Chapters:

Chapter Abstracts

read more

Publications

Published

In preparation

read more

Thesis Abstracts

Chapter 2: A hybrid approach for solving the gravitational N-body problem with Artificial Neural Networks

Abstract

Simulating the evolution of the gravitational N-body problem becomes extremely computationally expensive as N increases since the problem complexity scales quadratically with the number of bodies. In order to alleviate this problem, we study the use of Artificial Neural Networks (ANNs) to replace expensive parts of the integration of planetary systems. Neural networks that include physical knowledge have rapidly grown in popularity in the last few years, although few attempts have been made to use them to speed up the simulation of the motion of celestial bodies. For this purpose, we study the advantages and limitations of using Hamiltonian Neural Networks to replace computationally expensive parts of the numerical simulation of planetary systems, focusing on realistic configurations found in astrophysics. We compare the results of the numerical integration of a planetary system with asteroids with those obtained by a Hamiltonian Neural Network and a conventional Deep Neural Network, with special attention to understanding the challenges of this specific problem. Due to the non-linear nature of the gravitational equations of motion, errors in the integration propagate, which may lead to divergence from the reference solution. To increase the robustness of a method that uses neural networks, we propose a hybrid integrator that evaluates the prediction of the network and replaces it with the numerical solution if considered inaccurate.

Hamiltonian Neural Networks can make predictions that resemble the behavior of symplectic integrators but are challenging to train and in our case fail when the inputs differ ~7 orders of magnitude. In contrast, Deep Neural Networks are easy to train but fail to conserve energy, leading to fast divergence from the reference solution. The hybrid integrator designed to include the neural networks increases the reliability of the method and prevents large energy errors without increasing the computing cost significantly. For the problem at hand, the use of neural networks results in faster simulations when the number of asteroids is ~70.

Chapter 3: A Generalized Framework of Neural Networks for Hamiltonian Systems

Abstract

When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure may be crucial for many problems. At the same time, solving chaotic or stiff problems requires integrators to approximate the trajectories with extreme precision. Integrating Hamilton’s equations to a level of scientific reliability such that the answer can be used for scientific interpretation, may lead to computationally expensive simulations. In some cases, a neural network can be a viable alternative to numerical integrators, offering high-fidelity solutions orders of magnitudes faster.

To understand the role of preservation of symplecticity in problems where neural networks are used, we analyze three well-known neural network architectures that include the symplectic structure inside the neural network’s topology. Between these neural network architectures, many similarities can be found. This allows us to formulate a new, generalized framework for these architectures. In the generalized framework Symplectic Recurrent Neural Networks, SympNets, and HénonNets are special cases. Additionally, this new framework enables us to find novel neural network topologies by transitioning between the established ones.

We compare new Generalized Hamiltonian Neural Networks (GHNNs) against the already established SympNets, HénonNets, and physics-unaware multilayer perceptrons. This comparison is performed for the gravitational three-body problem. In order to perform a fair comparison, the hyperparameters of the different neural networks are chosen such that the prediction speeds of all four architectures are the same during inference. A special focus lies on the capability of the neural networks to generalize outside the training data. The GHNNs outperform all other neural network architectures for the problem considered.

Chapter 4: Reinforcement Learning for Adaptive Time-Stepping in the Chaotic Gravitational Three-Body Problem

Abstract

Many problems in astrophysics cover multiple orders of magnitude in spatial and temporal scales. While simulating systems that experience rapid changes in these conditions, it is essential to adapt the (time-) step size to capture the behavior of the system during those rapid changes and use a less accurate time step at other, less demanding, moments. We encounter three problems with traditional methods. Firstly, making such changes requires expert knowledge of the astrophysics as well as of the details of the numerical implementation. Secondly, some parameters that determine the time-step size are fixed throughout the simulation, which means that they do not adapt to the rapidly changing conditions of the problem. Lastly, we would like the choice of time-step size to balance accuracy and computation effort. We address these challenges with Reinforcement Learning by training it to select the time-step size dynamically. We use the integration of a system of three equal-mass bodies that move due to their mutual gravity as an example of its application. With our method, the selected integration parameter adapts to the specific requirements of the problem, both in terms of computation time and accuracy while eliminating the expert knowledge needed to set up these simulations. Our method produces results competitive to existing methods and improve the results found with the most commonly-used values of time-step parameter. This method can be applied to other integrators without further retraining. We show that this extrapolation works for variable time-step integrators but does not perform to the desired accuracy for fixed time-step integrators.

Chapter 5: Reinforcement Learning for the determination of the bridge time step in cluster dynamics simulations

Abstract Astrophysical simulations often deal with multi-scale and multi-physics cases, which entails the need for methods which can handle those. A common approach involves separating the system into multiple parts where each is integrated using problem-specific methods. Then, those parts are coupled on a given time scale. The coupling time scale, or coupling integration time, is determined manually and remains fixed throughout the simulation. In this work, we introduce a novel approach that leverages reinforcement learning techniques to automatically select the coupling time step during simulations. Our method effectively balances computation time and accuracy by adapting the time-step size to the characteristics of the simulation at each step. We test our method on a multi-scale problem: a star cluster in which one star contains a planetary system. We perform multiple tests on clusters with different (small) numbers of bodies and find that the method remains robust for multiple cases and across multiple physical domains. We test the effect of changing the integrators used for the different parts of the system and demonstrate that our method is independent of this choice. Similarly, we implement a parameter that scales the time steps to tune the accuracy requirements of the problem and find that our method is also applicable in this case. For long-term integration, where energy errors tend to accumulate, we find that our reinforcement learning method can achieve better results than the methods with fixed-time steps, but all cases lead to large energy errors in time. We develop a hybrid strategy that can detect jumps in energy error and prevent them by recursively reducing the time-step size at a given instance. The case with the hybrid implementation performs orders of magnitude better compared to our baselines, without significantly increasing the computation time. This method ensures the robustness of simulations that use reinforcement learning.

Our method eliminates the need for expert knowledge and balances computation time and accuracy while adapting to the needs of the simulation at each step. We show that it can be directly extrapolated to a large range of astrophysical simulations.

read more

Reinforcement Learning for the simulation of a star cluster

Reinforcement Learning for the Determination of the Bridge Time Step in Cluster Dynamics Simulations

The three-body problem is famously complex. As there is no analytical solution to predict the future state of the system, we rely on numerical simulations. These simulations are approximations, and their accuracy depends on certain simulation parameters. An important one is the time-step size. The smaller this value, the more the integration approximates a continuous solution, and the more accurate the simulation will be. However, reducing the time-step size leads to a larger computational cost of the simulation.

Chapter 3 Chapter 3

We want to balance accuracy and computational cost to achieve efficient simulations that still allow us to extract scientific conclusions. To do that, we develop a reinforcement learning algorithm that automatically selects it for you. By doing so, we also allow this time-step parameter to change and adapt to the needs of the simulation to keep the accuracy requirements. Our method achieves better results than any of the current methods.

The code is publicly available at Github link

See Publication

read more

Reinforcement Learning for the three-body problem

Reinforcement Learning for Adaptive Time-Stepping in the Chaotic Gravitational Three-Body Problem

We introduced the idea of using reinforcement learning algorithms to choose essential simulation parameters automatically. Here, we extend the idea to a more complex case, including multi-scale astrophysical systems. We create a method that balances accuracy and computation time while achieving better results than the current methods. Additionally, we ensure the robustness of the method for long simulations by creating a hybrid method that checks the quality of the reinforcement learning choices.

If you find it interesting, take a look at the Publication.

Also, the code and trained models are publicly available at Github link.

Chapter 4 Chapter 4

read more

ESA Alpbach Summer School: design of a Space mission to the Gas Giants

Alpbach Summer School: Design of a space mission to a gas Giant

The Saturnian system has been the target of four previous space missions and has generated high scientific interest due to its dynamic atmospheric properties and its many moons, some of which are key targets for the search for habitability. The SIREN mission is designed to study these environments, focusing on three main science goals. Firstly, the mission aims to study Saturn’s atmospheric composition to further understand its formation and evolution. Secondly, it aims to study the Saturnian ring and moon formation and evolution as a proxy for accretion processes in protoplanetary discs. Lastly, the mission will study what chemical processes shape Enceladus’ potentially habitable subsurface and surface environment.

Albach Schematic

The orbiter will perform multiple flybys of Enceladus at different altitudes crossing the plume to unambiguously characterise its chemical composition as a probe of this moon’s pristine interior material. After that, the orbiter will perform close observations of the A and B rings. The Saturn atmospheric entry probe (ARGO) will carry out measurements during a 90-minute descent to characterize the atmospheric composition down to an altitude of 20 bar.

See the Presentation slides and Report.

Post Alpbach: Concurrent Engineering design of a mission to Neptune

As a continuation of the Alpbach workshop, we continued the development of a space mission, this time to Neptune and its Moon, Triton.

This mission requires a complex trajectory, including a retrograde orbit around Neptune that allows for the mapping of Neptune’s atmosphere and magnetic fields. Then, the spacecraft is inserted into a Triton orbit to map its surface to the last details. It is important to take into account the current regulations for spacecraft disposal at the end of the mission.

Publication coming soon!

read more

Physics-aware neural networks for large astrophysics problems

A hybrid approach for solving the gravitational N-body problem with Artificial Neural Networks (2021-2022)

The numerical integration of a system of N-bodies, the calculation of the mutual forces between bodies can drive the computation time. The computational complexity scales quadratically with the number of bodies. This means that simulating a large system is… expensive.

Machine Learning can help with that. If instead of calculating these forces, we use neural networks to predict the total acceleration, we can save a lot of computing power. Physics-aware neural networks incorporate some of our physics knowledge into the neural network. In this case, we choose Hamiltonian Neural Networks (HNNs) and apply them to the integration of a planetary system with a large number of asteroids. We find that there are advantages and challenges to the application of HNNs to complex cases such as the gravitational N-body problem.

See Publication

read more

Optimization Comparison for a Earth-Mars trajectory

Optimization algorithm comparison of a spacecraft trajectory to Mars (2020)

A determining factor in the success of an interplanetary mission is the calculation of the mass of fuel that is needed. In order to increase the payload mass that can be carried, the mass of fuel must be reduced.

I compared the efficiency of different optimization methods to reduce the mass of fuel required for the transfer while ensuring the constraints are met.

Optimization Algorithms

See the full report with the explanation of the method and a comparison of optimization methods for a simplified version of the problem and a more complex one in Report.

read more

Master Thesis

Link to thesis

Abstract

The use of low-thrust propulsion for interplanetary missions requires the implementation of new methods for the preliminary design of their trajectories. This thesis proposes a method using the Monotonic Basin Hopping global optimization algorithm to find feasible trajectories with optimum use of the mass of fuel for the case in which the trajectory is modeled using the Sims-Flanagan transcription method. Due to the large computational time required to find the global optimum, Artificial Neural Networks have been used to predict the objective value and feasibility terms of the local minimum. Therefore, the procedure to set up a working regression Artificial Neural Network is studied as well as its transferability to predict values outside the trained limits and for different missions. In addition to this, the use of pre-training is analyzed to improve the performance of the network without increasing the size of the training database.

read more