A Windfall!

Dean Hu sat behind his desk, beaming as he waved Qi Hao in.

Old Wang sat off to one side, while several people in suits and ties occupied the sofa in the middle.

The middle-aged man leading them was the first to stand. He wasn't particularly tall or imposing, but he carried the air of someone in authority.

He held out his hand and said cheerfully, "You must be Qi Hao, right? You're even more handsome than you looked in your photo! What's going on? Does your school of mathematics admit students based on their looks?"

Qi Hao was taken aback for a moment, then quickly reached out to shake his hand. "You flatter me," he replied politely.

Dean Hu stood up at just the right moment. "Qi Hao, come, let me introduce everyone."

He gestured toward the middle-aged man. "This is CEO Lan of Huaji Technology Co., Ltd. In communications infrastructure, they're a pillar of our country!"

CEO Lan waved a hand and smiled. "Dean Hu, you're too kind. We engineers can't compare with the purity of you academics."

Dean Hu hardly dared agree. He immediately said, "CEO Lan, you're making us pen-pushers blush. If you weren't out there blazing trails and turning theory into tools that work in the real world, wouldn't the formulas in our hands all be castles in the air, nothing but battles fought on paper?"

"At most, we're foresters tending the nursery in the rear, watering these fine young saplings. But whether they can truly shine and give off heat for the country ultimately depends on sending them to hone their skills on a top platform like Huaji, where they have room to really spread their wings."

His words not only heaped praise on Huaji and CEO Lan but also complimented his own school's academic foundation. In the end, he had even steered the conversation unobtrusively back to the main topic of university-enterprise cooperation and supplying talent. It was undeniably smooth.

CEO Lan naturally understood what he meant. He laughed heartily and turned to the man beside him in a dark blue polo shirt. "Old Wang, did you hear that? I told you the University of Electronic Science and Technology was worth a visit!"

"Qi Hao, this is CEO Wang of Dafei Drone Company," Dean Hu hurriedly introduced him.

"Not long ago, CEO Wang led a team through a high-altitude drone flight test on the plateau. You could say they were the first company in our country to do it."

Dafei? Drones?

"Hello, CEO Wang." Qi Hao's interest was piqued. "High altitude, low air pressure, and strong wind shear on the plateau-to keep a drone flying steadily in that environment, wouldn't you have to redesign the attitude control algorithms from scratch?"

CEO Wang clearly hadn't expected Qi Hao to zero in on the key technical issue at once. A hint of surprise flashed in his eyes.

"You know about flight control too?"

Qi Hao shook his head and said frankly, "No, but I know that high altitude means a sharp drop in air density, which significantly reduces rotor lift. At the same time, strong winds place very high demands on the controller. These are engineering problems, but I think they count as mathematical problems too."

The corners of CEO Wang's mouth lifted slightly. He didn't reply, merely watching Qi Hao from the side.

Once everyone had sat down again, the conversation moved from pleasantries to business.

CEO Lan pulled a printout of a paper from his briefcase. The pages were covered in dense markings and notes, with at least three different colors of highlighter alone.

Qi Hao took a closer look. It was the paper he had written.

CEO Lan turned to page five and pointed to a theorem. "What about this asymptotic convergence theorem? Our technical team spent a long time discussing it. You proved that as the depth of the game tree approaches infinity, the search complexity of absolute pruning asymptotically converges to a lower bound."

"That lower bound happens to be O(b^{d/2}), the same efficiency as Alpha-Beta pruning with perfect ordering under ideal conditions?"

"More precisely, that applies to a class of game trees satisfying local consistency conditions. I gave a rigorous definition of local consistency in the paper. You can think of it as the evaluation function of a subtree satisfying Lipschitz continuity within a local neighborhood."

He took the pen CEO Lan handed him and wrote a line of equations in a blank space on the paper...

CEO Lan stared at the equation for a while, then flipped forward a few pages. "So your framework is currently a purely rule-based game search, right?"

Qi Hao nodded. "The entire paper discusses search optimization for deterministic, perfect-information games."

CEO Lan glanced at the technical specialist beside him, who immediately understood and took over. "Qi Hao, I'm the head of the search optimization group at the Huaji Research Institute. My surname is Chen. Our group read your paper line by line. The mathematical derivations are very elegant, and we have no objections to them, but we've run into a dilemma in practical applications."

"When we optimize the placement of communications base stations, the relationships among variables like terrain, obstructions, signal attenuation, and user distribution are extremely complex. We can't even write an exact evaluation function."

"Without an evaluation function, the premise of the pruning theory in your paper doesn't hold. So what we want to ask is, when the evaluation function itself is unknown, do you think there's still a way forward for search optimization?"

Qi Hao fell silent.

The question struck right at the boundary of his theoretical framework.

All his derivations rested on the assumption that the evaluation function was known. If there wasn't even an evaluation function, what was left to do?

"How are you handling it now?" Qi Hao didn't answer directly, instead putting a question to him first.

Engineer Chen gave a wry smile. "Exhaustive enumeration plus human experience. Engineers manually set some rules based on past projects, then brute-force search through the candidate solutions. It's very inefficient, and it relies heavily on the personal judgment of veteran engineers. When they leave, their experience goes with them."

"If you can't write the evaluation function... could you just not write it?" Qi Hao muttered.

CEO Lan raised an eyebrow. "What do you mean?"

Qi Hao weighed his words. "I mean, if the mapping between parameters and results is too complicated, and the cost of modeling it manually is too high or it's simply impossible, is there a way to let some mathematical structure approximate that unknown mapping on its own, directly from historical data?"

"Something like... using data to fit a universal function approximator?"

CEO Wang suddenly sat up straight on the sofa. "We've actually already been exploring that approach in the drone field."

Qi Hao turned to look at him.

"In drone flight control, the traditional solution is a PID controller, a purely mathematical model. It's very precise, but only if you have a complete physical model of the flight environment."

"But that approach doesn't work on the plateau. Wind-field turbulence patterns change in countless ways, and air pressure and density change nonlinearly with altitude. The model assumptions of classical control theory all fall apart."

"Later, one of our engineers, who also had a mathematics background, came up with a bold idea: feed the flight data straight in and let the system learn on its own."

"Learn?" Qi Hao immediately seized on the word.

"That's right, learn." CEO Wang nodded. "He built a multilayered mathematical structure. The inputs are the aircraft's current state parameters: attitude angles, angular velocity, air pressure, and wind speed. The outputs are speed commands for the four rotors. In between are several layers... how should I put it? You can think of them as layer upon layer of nonlinear transformations."

"Each layer performs a linear combination, then passes it through a nonlinear activation. There are more than ten thousand parameters, all adjusted using data."

Multilayered, nonlinear transformations, function approximation.

It was as if something in Qi Hao's mind had been pushed open a crack.

"Wait, you're saying that, in theory, this structure can approximate any continuous function?"

CEO Wang paused, then smiled. "Your intuition is spot on. There is a mathematical theorem, I think it's called the universal approximation theorem, that says just that. As long as there are enough nodes in the hidden layer, this kind of structure can approximate any continuous function to any degree of accuracy."

"Then for Huaji's base station placement problem, if you organized the historical placement data, used terrain parameters, user distribution, and obstruction data as inputs, and an assessment of signal coverage quality as the output, could you also use this structure to approximate that unknown evaluation function?"

Engineer Chen nodded. "The idea is right. In fact, people in the industry are already making similar attempts. Academia calls this field machine learning. The kind of multilayered structure you just mentioned is usually called a neural network."

"But our base station placement model has more than two hundred input dimensions. Later in training, the gradients fall almost to zero, and the model can't learn at all."

Qi Hao couldn't quite make sense of it when he heard this. He spotted a whiteboard nearby, walked over, picked up a pen, and began to write.

"Suppose the network has n layers, and the activation function of each layer is σ. During backpropagation, the gradient has to pass through the chain rule n times... for a ten-layer network, the gradient attenuation factor is 0.25 to the tenth power."

He wrote the result beside it: 0.25¹⁰ ≈ 0.0000009537.

Qi Hao put down the pen and turned around. "The gradient hasn't vanished. It's just decaying exponentially. This is a structural defect caused by the choice of activation function, not a problem with the optimization algorithm."

Old Wang watched from the side, nodding repeatedly.

What his student had said was indeed correct.

CEO Wang asked, "Then how do you solve it?"

"Change the activation function."

Engineer Chen stared at the broken line, his lips moving. "It's that simple?"

"The most beautiful things in mathematics are often simple. Of course, this is only my intuition. A rigorous proof of convergence will still take time, but from the perspective of gradient propagation, this direction should be right."

CEO Lan and CEO Wang exchanged a glance.

CEO Lan spoke first, no longer beating around the bush. "Qi Hao, I'll be direct. Huaji and Dafei have recently been discussing a joint project. Huaji would handle intelligent site selection and resource scheduling for ground base stations, while Dafei would handle autonomous flight control and flight-path planning for low-altitude drone relay nodes."

"The two lines of business look unrelated, but the core bottleneck is the same problem..."

"How to use a data-driven approach to replace human experience in complex environments where traditional mathematical modeling fails," Qi Hao finished for him.

CEO Lan nodded in agreement. "Right, exactly."

CEO Wang rose from the sofa, unusually serious. "I'll be honest. We at Dafei have engineering teams, and Huaji has data and computing power, but both sides lack mathematical talent."

"Now that we've met you, Qi Hao, we have some confidence."

"You can do more than mathematical derivations. You also have engineering intuition. Our engineers discovered the vanishing gradient problem too, but all their proposed solutions started from the engineering angle of tuning parameters."

CEO Lan nodded, then turned to Dean Hu. "Dean Hu, Huaji's technical committee approved a special budget just last week. If your school is willing, Huaji can contribute eight million yuan and Dafei four million. Together, the three parties can establish a Joint Laboratory for Intelligent Computing and Control at the school of mathematics."

"Twelve million yuan in initial funding. In addition, Huaji will directly supply all the hardware and high-end servers, while Dafei can also give the lab unconditional access to its latest prototypes and Mount Everest test data."

"But there's one condition: Qi Hao needs to take charge of our project."

Twelve million!

When it came to attracting funding, ten schools of mathematics couldn't beat a single school of computer science or communications.

Most of their usual funding came in sums of hundreds of thousands or a few million yuan. A collaboration windfall in the tens of millions was a rare sight!

He exchanged a glance with Old Wang.

The two of them looked calm on the surface, but their eyes were full of astonishment.

Holy shit!

Apparently, without either of them lifting a finger, Qi Hao had landed them a major project!

Old Wang had still been fretting about securing projects earlier, but he never expected a windfall to fall right into his lap today!