A genius? I just love studying.

Chapter 218 Human Strength Has Its Limits (Second Update)

Everyone in the lab sprang into action, and Chen Hui was no exception; he was already continuing to observe the experimental data.

They seemed completely unconcerned about the experimental results.

He didn't focus on the machine learning model issue. Professor E has already given him a brief introduction. They use geometric transformations and physical simulations to balance the sample distribution for defective samples.

Transfer learning was used to transfer the pre-trained silicon carbide defect detection model to gallium oxide. Meta-learning was employed to enable the model to quickly learn features such as the direction of line defects in dislocations from a small number of defect samples. This was combined with active learning to actively label "difficult samples" and guide engineers to supplement high-value data.

Some progress has already been made.

Academician E Weinan is an expert in machine learning, and Chen Hui did not interfere much, believing that these problems would be solved soon.

Chen Hui looked at another problem: in the mold-guided growth method, the melt flow affects the temperature gradient, which in turn causes crystal stress, resulting in an excessively high defect density in the final crystal, making it a waste product.

This problem involves the coupling of multiple physics fields, including temperature field, flow field, stress field, and electric field.

If only the temperature field is considered, the FEA model predicts a grain boundary defect density of 10 cm, but in reality, due to melt convection disturbance, the defect density reaches 3 × 10 cm.

Therefore, collaborative problem-solving is necessary to obtain more accurate data.

The temperature field involves the heat conduction equation, the flow field involves the Navier-Stokes equation, and the stress field involves the elasticity equation. These equations are already complex enough in their respective fields, and solving them together is undoubtedly even more difficult.

Of course, the solution here is not the same concept as the Navier-Stokes equation in mathematics. In mathematical research, the purpose of solving the Navier-Stokes equation is to explore the properties of the equation itself, such as the existence, uniqueness and smoothness of the solution.

This often involves complex mathematical analysis and proofs, such as using advanced mathematical tools like Banach's fixed-point theorem.

However, in practical applications, the main purpose of solving the Navier-Stokes equations is to predict and simulate the motion behavior of fluids. The goal is to find approximate numerical solutions that meet engineering accuracy requirements, without pursuing rigorous mathematical proofs.

It can be said that it is not closely related to the problem that Chen Hui is researching, but there is no doubt that if Chen Hui can complete the proof of the mathematical solution of the Navier-Stokes equations, it will bring huge benefits to the approximate solution in engineering applications, and may even bring about an epoch-making revolution.

Gathering his thoughts and returning to the problem at hand, Chen Hui reviewed the entire experimental process in his mind. The Navier-Stokes equations describe the conservation of momentum in fluids, the heat conduction equations describe energy transport, and the elasticity equations describe the stress-strain relationship. However, in the process of gallium oxide wafer growth, the three are strongly coupled and cannot be solved independently.

The flow of melt causes a temperature gradient through convective heat transfer, which changes the local coefficient of thermal expansion, induces thermal stress, and generates coupling from the flow field to the temperature field.

Uneven temperature leads to differences in thermal expansion/contraction of materials. During wafer growth, thermal stress is generated at the interface between the seed crystal and the melt, and the temperature field in turn affects the stress field.

During crystal growth, stress near the solid-liquid interface may change the melt viscosity, stress field around dislocations may affect the diffusion coefficient, and may even induce flow disturbances such as centrifugal force when the crystal rotates. The stress field may also affect the flow field.

This "two-way strong correlation" means that traditional single-field solvers cannot be directly extended and need to handle the cross-coupling of nonlinear terms, such as the viscous coupling between viscous stress and stress field in the Navier-Stokes equations.

During gallium oxide melt growth, thermal stress may cause fluctuations on the melt surface, which in turn changes the shape of the melt-gas interface and affects the gas protection effect (e.g., fluctuations in oxygen content), forming a multi-level coupling of "flow field-temperature field-stress field-chemical field".

The three are interdependent. To make an accurate prediction, the three equations must be solved simultaneously. This means that data exchange is required for each step of the solution process. Only by nesting the solutions in this way can the accuracy of the final prediction be improved.

However, the three equations were already complex enough, and with such nesting, the difficulty of solving them increased exponentially. Furthermore, the data in each field would change during the data exchange process, making it difficult to obtain accurate predictions. This is why Professor E and his team have not been able to solve the problem yet.

This problem can be solved entirely by occasional loose-tight co-operation, which reduces the difficulty of solving the problem, reduces the amount of computation, and reduces data delay caused by data exchange. Chen Hui began to deduce the solution on the draft paper.

Loose coupling refers to solving each field independently, reducing computational load through low-frequency data exchange, such as updating the temperature field every 10 steps of flow field calculation. It can be used in scenarios with weak coupling, such as in the later stages of steady-state growth when the flow field and temperature field have become stable.

Tight coupling employs "nested iteration" or "unified time step," iterating the field equations multiple times within each global time step until the residuals meet the accuracy requirements. This approach is suitable for strongly coupled scenarios, such as the early stages of growth when melt flow is intense and temperature gradients are large.

The loose-tight coordination method is commonly used in engineering. The reason why Academician E Weinan and his colleagues have not yet solved it is simply because they have not been able to design an inter-field data transmission interface and have not found an efficient convergence criterion.

Coincidentally, these are all things that Chen Hui excels at.

However, this problem has not been completely solved. To achieve tight and occasional synchronous nesting, it is necessary to unify the spatiotemporal scale of each field. The difficulty lies in the significant differences in the characteristic spatiotemporal scales of different physical fields.

The characteristic timescale of melt flow is in the millisecond range. In the guide mold method, the melt flow velocity is about 0.1 m/s, the characteristic length is 0.1 mm, and the timescale τ≈1ms.

The characteristic timescale of thermal diffusion is on the order of seconds. With a thermal diffusivity of approximately 10^6 m2/s for gallium oxide and a characteristic length of 1 mm, the timescale τ≈1 s.

The characteristic timescale of crystal growth is on the order of hours, with an 8-inch wafer growth cycle of about 2 hours. However, the timescale of stress relaxation can be as short as minutes, such as the timescale of dislocation movement.

This "timescale separation" means that the traditional global time step cannot balance accuracy and efficiency. If the time step of the flow field is taken as 1ms, the temperature field and stress field need to be calculated 1000 times/second, resulting in an explosive computational load. If the time step of the stress field is taken as 1 minute, the transient effects of the flow field and the unsteadiness of the flow initiation stage will be ignored.

The busy work in the laboratory has come to an end. Next, it's time to observe the parameters of each step in the experimental reaction and verify the accuracy of Chen Hui's model.

Zhang Xing looked at Chen Hui, who was deep in thought on the control panel, and felt a mix of emotions. They admired Chen Hui a lot; he could enter a deep learning state anytime, anywhere, which was a talent that many scholars dreamed of. They also understood why Chen Hui had achieved so many remarkable accomplishments.

But this guy managed to tinker with a model in just over ten minutes, and when they asked him to verify it, it felt a bit like child's play.

Hey!
Chen Hui sighed softly and rubbed his slightly sore neck.

"Teacher, how about we take a break?"

Chloe asked with some concern, "The teacher has been lying on the control panel for more than two hours. Even an iron man couldn't withstand that."

Chen Hui waved his hand. He felt he had some inspiration, but his mind was a mess and the inspiration was not clear.

"Choi?"

"International student?"

"Teaching according to aptitude!"

Suddenly, upon seeing Chloe, these words flashed into Chen Hui's mind, like a withered tree struck by lightning, leaving only a charred branch, but beneath that branch, a tender bud quietly grew.

Since the spatiotemporal scales of various physical fields are not the same, why must we use the same time step?
To address the timescale separation problem, a multi-time-step strategy can be adopted, setting the global time step to the characteristic time Δtflow of the flow field. Within each global step:

1. The flow field directly propels Δtflow;

2. The temperature field is advanced with a smaller time step (Δttemp=Δtflow/n, n=10100), and the results of the flow field are used as boundary conditions;
3. The stress field is advanced with smaller time steps (Δtstress=Δtflow/m, m=1001000), and fine calculations are performed only in high stress gradient regions, such as the solid-liquid interface, while quasi-static approximations are used in other regions.

By introducing the "local time step" technique, a small time step is used only in areas requiring high precision (such as near the interface), while a large time step is used in other areas, thus balancing efficiency and accuracy!
Isn't this another form of individualized education?

Once he figured out the key issue, everything became clear, and Chen Hui quickly began to work out the solutions on the draft paper.

"It's done!"

“It really worked!”

Suddenly, a burst of joyful cheers erupted in the laboratory.

Yang Chi, Zhang Xing, and others who were conducting the experimental verification were so excited that they danced with joy, as this was the only way to express their excitement.

By calculating the formation energies of Ga and O using DFT and correcting the diffusion coefficient formula of the LSW model, the prediction error of the melt growth rate using the model provided by Chen Hui was reduced from 25% to 5%!

They just conducted ten sets of experiments, and the error of each set was between 4% and 7%.

Zhang Xing couldn't believe his eyes. The guy on the control panel had achieved this result in just over ten minutes. What had they been doing for the past two months?
This is a model that Chen Hui roughly calculated based on previous data. If online sensors, infrared thermal imagers, gas chromatographs, etc. are deployed on the production line to collect temperature and gas composition data in real time, and the model parameters are dynamically corrected through Bayesian optimization algorithms, this error data will be further reduced.

When Zhang Xing looked at Chen Hui again, his eyes had changed.

This guy is amazing!
Not only Zhang Xing, but all the researchers who had just joined the lab were shocked.

Even though they had already heard about Chen Hui's many achievements and knew that he was a genius with profound knowledge of mathematics.

But no matter how many stories you hear, none of them have the same impact as a story happening right before your eyes.

Not to mention them, even Yang Chi, Deng Ting and others were excited. They knew that Chen Hui's arrival would definitely bring about something and a miracle, but even they did not expect that this miracle would come so quickly.

E Weinan also sighed, looking at the young face beside him. It's true that you can't deny getting old.

Chloe, standing behind Chen Hui, already had stars in her eyes. The already divine figure seemed even taller in her eyes. She would believe it even if someone told her that Chen Hui was the reincarnation of Jesus.

Only Chen Hui didn't show much reaction, which was exactly what he had expected.

"nailed it!"

It took Chen Hui nearly another hour to complete the multi-scale spatiotemporal reduction and layering operations, and to put together an applicable model.

The validation of this model is more complex; it requires first converting the model into a computer model before validation.

"Great!"

E Weinan looked at the draft paper that Chen Hui handed him with some excitement.

He called Chen Hui over because he wanted Chen Hui to help solve the problem of multi-field coupling. As long as a prediction model that meets the accuracy requirements can be built, the difficulty of controlling various parameters in the gallium oxide wafer growth process will be greatly reduced. If the hardware accuracy is insufficient, software prediction can make up for it.

As for the correction of the solution diffusion model, it can only be considered an unexpected bonus.

Now, with both factors combined, as long as Chen Hui's prediction model doesn't deviate too much, the industrialization of gallium oxide will come naturally.

But Chen Hui was not satisfied.

His chemistry proficiency was close to level 3, although it was still far behind many materials science professors. However, his math skills were good enough, and his early experience in the lab made him seem to have divine assistance when solving these materials science problems.

Once in the lab, the problem can be easily identified.

Another key issue is that in gallium oxide wafer growth, the interfaces of multiphysics fields, such as the melt-crystal interface, crucible-melt interface, and gas-melt interface, are key coupling regions, but their characteristics are difficult to describe precisely.

The surface tension σ and fluctuations exist at the free surface melt-gas interface. The interface needs to be tracked using the VOF or Level Set method, while simultaneously coupling the surface tension term ((σκ) of the NS equation.

At the solid-liquid interface crystal-melt interface, there is latent heat of phase transition during gallium oxide crystallization, so the latent heat term ρLtf needs to be added to the heat conduction equation. At the same time, the stress continuity at the interface (σsolid=σliquid) must be strictly satisfied.

In actual processes, random factors such as the temperature distribution on the crucible wall, fluctuations in heater power, and pulsations in gas flow introduce uncertainties in boundary conditions, which traditional deterministic solvers struggle to capture in terms of their impact on multi-field coupling.

E Weinan could naturally understand what Chen Hui was doing. His eyes swept around the laboratory, and the jubilation subsided wherever his gaze fell. This was partly due to the powerful pressure from Elder E, and partly because they knew they couldn't disturb Chen Hui at this time.

But they didn't just stand by and watch. E Weinan had already picked up the draft paper next to Chen Hui and started directing everyone to input the model for verification.

This time, Chen Hui fell into deep thought.

Identifying the problem is half the battle won, but completely resolving it is clearly not so easy.

Half an hour passed, and he still hadn't found any useful ideas.

call!
After another ten minutes or so, Chen Hui let out a long sigh. He finally raised his head, rubbed his throbbing head, and decided to put the problem aside for the time being until his chemistry and math grades improved.

If you have no clue about a problem, it means it's not something that can be solved in a short time. At this point, you need to temporarily put the problem aside, keep learning and improving yourself, until things fall into place and the problem is solved.

Chen Hui is very experienced.

(End of this chapter)