These trees love water, and they have very thirsty roots. It is for this reason that they are so strongly associated with ponds and are often found growing near bodies of water. As a non-native species, it is quite possible that many weeping willows you have seen were deliberately planted, rather than propagating through natural means. They can be quite useful when it comes to strengthening shorelines, as their roots form extensive networks and may seem drawn to water like a magnet. This can help hold a sandy lakeside together and prevent it from being washed away into the water during storms.
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So simply digging up the roots and trimming them way back from the pond is not going to eliminate the likelihood of the tree just regrowing them back and piercing your fresh liner like they did the first time.
What about a barrier, you may be wondering. A solid object that the roots cannot penetrate should keep our liner safe from the tenacious roots of the hydrophilic willow, shouldn't it? Perhaps something like concrete cloth...
Well, you are thinking in the right direction, but I would not use concrete cloth. Concrete will wick water through it, attracting thirsty roots. Not to mention that it is in the nature of concrete to crack. After all, that is why expansion joints are used when pouring slabs and sidewalks. The joints are to confine and control where the cracking occurs. Even if you can't see them, there are still going to be microscopic cracks in it. Don't think you can just make the concrete super thick, either. Even at 660' thick, Hoover Dam still leaks water! Luckily, there are no willow trees on the other side of the dam, because microscopic root hairs can enter such cracks and eventually penetrate the concrete and crack it further.
A better solution is to buy a root barrier. They come in panels that slide together and come in different sizes. The panels attach together so you can make the barrier as large or small as you want it. When it is finished, it is like you planted the tree in a plastic container with no bottom. The idea is to have the roots grow horizontally until they hit the impenetrable barrier, then grow down.
When doing this, it is also wise to dig a few vertical holes that are deeper than the bottom of the root barrier. The holes go between the barrier and the trunk of the tree. Sink a 4" pipe in them and fill them with gravel to allow deep water penetration, then install a bubbler or drip-emitter in there so the water can go deep into the ground; the roots will follow. It is important to water heavily, but not too frequently.