In the winter of 2004, Cai Guoqiang saw a photo of a submarine, made out of scrap metal and shaped like a fish, that had been built by a peasant (the artist's word) in landlocked Hubei province.

Cai, who was born in Fujian province in 1957, is one of China's most successful contemporary artists. He's probably most famous for his fascination with gunpowder; his "explosion events" have taken place worldwide. He has a particular eye for communal spectacle and, in 2008, he was director of visual and special effects for the opening and closing ceremonies of the Beijing Olympics. He has lived in New York since 1995. For a man who loves to test himself publicly by inventing difficult but fleetingly beautiful urban effects, the city is a good match.

Still, he yearns to stay close to what remains of China's rural heart. When he saw that photo of the submarine he contacted its creator, Li Yuming, and the pair met during the following Lunar New Year. Cai had already heard about such men - always men - who toiled over unlikely inventions in China's impoverished villages to little public acclaim. In Li's case, there had been problems with actually testing the submarine's underwater abilities and, eventually, local officials had instructed him not to continue. (He might, they worried, obstruct the flow of the river.)

Cai admits that his initial interest was prompted by amusement. But when he saw the submarine, named Twilight No1, he decided to buy it. He didn't know then that it would be the first component in a strange menagerie of the imagination he would source across China: wooden aeroplanes, spindly helicopters, robots, flying saucers, an aircraft carrier …

That last item was created by Tao Xiangli, another submarine inventor, who lives in Anhui, another landlocked province. Not having sufficient space at home, Tao was obliged to park his first submarine, concocted out of oil barrels and a wok, in a nearby reservoir, where it froze into the ice. When Cai came to visit and suggested that Tao might like to design an aircraft carrier, the latter agreed that China could certainly do with one. (This was before the People's Liberation Army unveiled the Liaoning, in 2012.)

See also: Farmer-cum-inventor - one of China's 'peasant Da Vincis' - enjoys life in his orange submarine

Cai, who has a good sense of humour, expressed a desire that his commissioned aircraft carrier should be better than American ones, to which Tao responded by slinging a couple of submarines - actually propane tanks - onto the sides. The result, titled Complex, was the 20-metre-long centrepiece of a 2010 exhibition curated by Cai, featuring the work of about 50 rural inventors. It was the inaugural event of Shanghai's Rockbund Art Museum, and took place during that city's World Expo.

The Expo's official slogan was "Better City, Better Life". Cai took those words and recast them for his show as "Peasants - Making a Better City, a Better Life". He wanted to bring peasants, who, after all, as migrant workers had built China's shiny economic miracle, to the forefront of people's appreciation.

He had also come up with a memorable collective term for the inventors: peasant da Vincis. After the show closed in Shanghai, it travelled to three cities in Brazil - Brasilia, Sao Paulo and Rio de Janeiro - under the title "Da Vincis do Povo" (" do povo" means "of the people" in Portuguese. Overseas liberals, unacquainted with its complimentary value in China, tend to wince at the word "peasants"). A million visitors went to see the inventions; according to The Art Newspaper, the Rio de Janeiro show was the world's most-visited contemporary exhibition in 2013.

Now the peasant da Vincis' inventions have arrived at Milan's National Museum of Science and Technology, in Italy. Cai's work is always site-specific and this location is doubly significant. Until October 31, Milan is hosting the World Expo; and it was also the city where the original da Vinci - that would be Leonardo, the Renaissance genius who made no distinction between art and science - spent much of his highly productive professional life.

ONE RECENT MONDAY afternoon, when the museum is closed, Mariluz Hoyos, head of projects at Cai's studio, stands gazing down into its courtyard, where a group of men has carefully placed Tao's aircraft carrier, vertically, into the earth. In Shanghai, it was displayed horizontally but the Milan museum, formerly a monastery, can't accommodate such a generous layout. As a result, it looks as if the garden has been pierced by a gigantic, rusty scalpel.

Below her, Tatsumi Masatoshi, Cai's technical director, who has worked with the artist since Cai lived for almost a decade in Japan from the mid- 1980s, is dealing with the Italian crew. It's been two years since the inventions have been seen in public; in the interim they've been packed away in Quanzhou, Cai's hometown in Fujian, where, one day, he'd like to have his own museum. Now, 10 days before the opening of the exhibition, Masatoshi and his assistants have to enter the makers' minds and piece the creations back together again.

"The guys say it's like solving a puzzle," says Hoyos. "These machines come without manuals so it's like reinventing them and it tells you about the thought processes of the inventors - you think, 'Wow, it's pretty ingenious.'" Glancing down at Masatoshi, she adds, "He's dedicated to making incredible things happen."

The most recent incredible thing was the Sky Ladder, a stupendous pyrotechnical feat that Cai realised with Masatoshi's help in June, and has been viewed 50 million times on Facebook. For 2½ minutes before dawn, the sky above Huiyu Island Harbour, in Quanzhou, burned with golden steps as if some invisible deity were ascending to the heavens. This was the fourth attempt and most perfect realisation of a fire-ladder dream Cai had had since childhood, since the days when he'd watched his father, a calligrapher who ran a government bookshop in Quanzhou, paint landscapes in a matchbox.

Cai went on to train in stage design at the Shanghai Theatre Academy, and his work is of a highly dramatic visual quality: apart from his experiments with gunpowder, he has suspended neon-radiating cars within New York's Guggenheim museum, assembled (and disassembled) various boats, and created extraordinary dioramas often featuring animals and water. His imagination is that of a child for whom art could be both tiny scenes in a matchbox and massive portraits of Mao Zedong carved into the mountains. (Guoqiang literally means "big country": his father was Maoist.) He's an inventor, too.

At the back of the Milan museum - Hoyos cuts through a huge shed filled with dark, silent steam trains - a row of inventions from China's heartland is laid out in the afternoon sun, next to a 1957 Italian fighter jet, waiting for Cai to assign them their places in the show. Li's Twilight Nos 1, 5, 6 and 7 are here. So are Du Wenda's Flying Saucers C, D, E and F, trembling a little in the afternoon breeze. (As the inventors don't consider themselves artists, they don't tend to indulge in fancy conceptual titles.)

Du is also from Anhui, where he founded the Global UFO Scientific and Technological Research Institute of Xiao County. It's the second time his inventions have been in Italy: one of his UFOs, a confection of stainless steel and propeller blades, took part in the 2005 Venice Biennale, when Cai was curator of the China Pavilion. On that occasion, Cai asked Du - as he climbed into his machine - how he planned to descend, to which Du replied, "Never learned how to land!" This blithe statement was made into a banner and hung on the façade of the Rockbund Art Museum.

In fact, that UFO caught fire and never took off. Du survived (in a children's guide to the Milan exhibition, there's a photo of Cai with a speech bubble assuring Du that none of Leonardo's inventions was able to fly, either) but not all of the inventors have lived to tell the tale. Yi Ruilong's hang glider is in the exhibition; while it was fluttering in the Sao Paulo show in February 2013, its creator, aged 70, died in another of his flying inventions, falling out of the sky, like Icarus, above Sichuan province's Hanyuan Lake.

And maybe this is the exhibition's attraction: it tells the eternal tale of mankind's yearning. These creations, gathered from places far from Brazil or Italy, have a universal poignancy that makes you simultaneously smile and sigh. Standing next to a wooden aircraft that looks as if it were painfully hewn out of balsa wood, Hoyos, who is Colombian, is asked to speculate on the state of mind of the inventor (Chen Zhongzhi, of Fujian province).

"I think he suffers," she says, quietly. "Someone sitting in there, staring at the horizon … it's poetic but so melancholic."

If there's any melancholy within Cai's own psyche, he keeps it well hidden. He's an extraordinarily good-natured and energetic man. Wherever he is in the world, he exercises morning and evening: his life, his art, the people he meets, all vary so much that he needs a physical constant. He's also so tall that his head didn't fit when he climbed into Tao's submarine.

"One point eight something," he says, vaguely, in Putonghua, when asked his height. (Despite his years in New York, he speaks no English, and Béatrice Grenier, from his studio, interprets. Cai, who has two daughters, is noticeably comfortable around women.)

As he walks through the museum the following day, planning his installation - the moment when he transmutes science into art - his presence is that of both impresario and alchemist. He's the enabler. High above him, on the museum's external wall, is an enormous billboard featuring an astronaut's helmet under the declaration "SPAZIO Adesso Puoi Avere La Luna!" ("SPACE and now you can have the moon!").

Six of the exhibition's inventors created flying machines. For three of them, travelling to Shanghai in 2010 was the first time they'd actually been on a plane. Cai had hesitated about inflicting that reality on them. He thought it might, somehow, disturb their vision but he needn't have worried; if anything, the experience reinforced the dream. The inventors told him that while their own craft properly resembled birds, a commercial flight was like taking a bus. They felt, they said, that they weren't truly flying.

Most of them have never heard of Leonardo da Vinci, although da Vinci, the illegitimate son of a 15th-century peasant mother, could have surely identified with their lives. When Cai, for instance, went to visit Wu Shuzai, who lives in the mountains of Jiangxi province, he found a family who had nothing - the floor of their home was the good earth. And yet Wu had felt compelled to stitch together a helicopter from parts of a threshing machine and chunks of wood. (His wife chopped up an early prototype for fuel because the family was so impoverished.) After Cai bought it, the village treated its departure - naturally, it couldn't fly so it was borne down the hillside on human shoulders - as a religious ceremony.

He is, Cai says, moved by beautiful pieces that he can't make himself. The word "beauty" might seem odd but it is apt; human chemistry makes these experiments more lovely the longer you stare at them, the more you hear of their birth pangs. And there's an entire collection of robots by Wu Yulu, who lives outside Beijing. They have a vivid, cheering presence, not least because some of them, commissioned by Cai, paint in the style of Jackson Pollock, Damien Hirst and Yves Klein.

Cai has been obliged to buy the works because the peasants refuse to lend them.

"They don't believe in this exhibition culture," he says. "They're afraid that people might trick them so they prefer to sell them. And they need the money - they know their inventions are not perfect so, with money, they can make them better."

Although perhaps his influence, and his money, may make subsequent commissioned pieces a little less charming … "selling can help them realise their dreams," he insists. "And it makes them confident in their villages. All day long there is gossip about them, and they feel useless. But their life is invested in this work."

He says he can't remember how much Twilight No1 cost. Do the peasants bargain?

"Not really. They are pure. Their common personality trait is that they're a little naïve."

He didn't think, at first, that such oddities could become part of his own art process. It was the Shanghai World Expo that made him realise he could tell a site-specific story about these objects.

"The peasants are expressing their value within society but our attention towards them is not adequate."

Did anything change after the expo?

"Objectively, the change is not very big," he replies, honestly. Some of the inventors thought of producing works for sale but all of them need more money - more sources of work - to develop their vision.

A documentary on Cai, by Kevin Macdonald - who directed The Last King of Scotland, Touching the Void and, in 2000, won an Academy Award for his documentary One Day in September - is currently in post-production. (The producers are Fisher Stevens who, in 2010, won an Academy Award for his documentary The Cove, and Wendi Deng Murdoch, Rupert's former wife.) The peasant da Vincis will not feature in the programme but seeing their work after a two-year gap is encouraging Cai to think of returning to China's villages and making a film specifically about them.

"I brought up these inventors' children," he says, "and I use my art to tell their story."

The recent death of his grandmother, 10 days after Sky Ladder, which he'd created for her 100th birthday, is also prompting thoughts of the motherland. As is the way with ladders, he can't decide if that feat was a beginning or an end. But he recognises his overlap with the peasants: "Both of us have to create a time-tunnel to travel to another world".

What will the peasant da Vincis' tales mean to the Italians? On the evening of the opening, the museum's garden is hung with submarines, planes, a helicopter dangling around Tao's upended aircraft carrier. Yi's paraglider is glimmering in the twilight. None of the inventors is here but perched high on one of the gantries is a robot, looking as if it's about to take exuberant flight.

The museum wanted the exhibition's slogans to be expressed in Chinese characters but for Cai, the point is not to be inexplicably exotic but to help people understand these men's aspirations; and so "Never Learned How to Land" is written in Italian (" Non ho mai imparare ad atterrare") on a banner.

Inside, among more peasant creations, the robots stand silent until Hoyos galvanises them. The Jackson Pollock robot begins its spattering of paper. A rickshaw-pulling robot starts moving. "Wu Yulu is my dad" is written, in English, on its back. The guests (talking in Italian about China's economy), the security guards, a group of men walking past the windows outside, all stop to stare. It's magic: a night at the museum, come to life.

Ten minutes' walk away is The Last Supper, the Leonardo da Vinci work that every tourist in Milan tries to see. Hardly anything of what da Vinci painted actually remains; it's as if the creator's soul has long gone. But facing it, on the opposite wall of the monastery refectory, is a huge fresco of the Crucifixion, painted in 1495, that's still vivid with humanity. You imagine the toil that went into it more than five centuries ago. The artist's name isn't famous and the visitors barely glance at it.

Cai Guoqiang: Peasant da Vincis continues at the National Museum of Science and Technology, Via San Vittore 21, Milan, Italy until January 6.