"What the president has asked us to do is turn over a new leaf to transform our relationship with the people of Iran," Vance said in the mountain resort of Burgenstock, where the talks kicked off just hours earlier on Sunday.
"We've already made great progress over just the last few hours, and I expect that we'll make additional progress," Vance said.
The talks - dubbed the "Lake Lucerne Summit" - began on Sunday afternoon on the shore of Lake Lucerne.
According to the foreign ministry of Qatar, which is mediating between the warring countries alongside Pakistan, the initial high-level meeting involved representatives from the US and Iran as well as delegations from Pakistan and Qatar.
Qatar said there was hope that the meetings would lead to a "comprehensive and lasting agreement on all points within the framework agreement," referring to a memorandum of understanding signed earlier this week by the United States and Iran that paved the way for the talks in Switzerland.
Under the terms of the memorandum, a final agreement on ending hostilities, as well as on Iran's nuclear program, is to be reached within 60 days.
However, the talks have been overshadowed by ongoing clashes between Israeli forces and the Hezbollah armed group in Lebanon, despite a ceasefire that is part of the memorandum.
"Iran must immediately stop their highly paid PROXIES in Lebanon from causing trouble. If they don't, we'll hit Iran very hard again, just like we did last week, only harder!!!" US President Donald Trump said in an online post in apparent reference to an escalation he ordered earlier this month.
At the talks, Vance played down the impact of violence in Lebanon, saying progress had been made towards ending hostilities there in recent days.
"These things are always a little bit messy," he said.
Despite the announcement of a new ceasefire in Lebanon on Friday, there has been scant sign of an end to fighting there.
Iran's military said on Saturday that as a result, it had again shut the Strait of Hormuz, whose closure for nearly four months caused the biggest disruption of global energy supplies in history.
US officials disputed whether the strait was again shut but commercially available shipping data showed an immediate effect.
Only a single small tanker crossed the strait with its location-signalling transponders on after Iran's announcement, compared with dozens of ships in recent days when traffic had begun returning to pre-war levels.
Earlier on Sunday Vance briefly appeared in front of travelling media as he, envoy Steve Witkoff and Trump's son-in-law Jared Kushner met Pakistani Prime Minister Shehbaz Sharif and Pakistan's Army Chief Field Marshal Asim Munir.
Warm greetings were exchanged, and Vance said "What's up man?" as he shook hands with Munir and hugged him.
"My brother," Munir said as he reached out to Witkoff and embraced him.
Iranian foreign ministry spokesman Esmail Baghaei said Sunday's talks would last for just one day.
Since the US had failed to guarantee a ceasefire in Lebanon, the talks would cover only the implementation of the memorandum itself and not the substantive issues foreseen for the next stage of negotiations, Baghaei said.
with DPA and AP
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