Thanks for this really excellent example. Minor correction in your code. The ("Time"->1) at the bottom of the "pde' net should be changed to ("Time" -> "Scalar") because of the input data:

Take[data,5]
{0. -> {970., 20., 0.}, 0.1 -> {969.572, 20.1854, 0.18824}, 
 0.2 -> {969.14, 20.372, 0.377969}, 
 0.3 -> {968.704, 20.5599, 0.569198}, 
 0.4 -> {968.266, 20.7491, 0.76194}}

Could you please explain why #[[1]] is used for "Sres", "Ires" and "Rres"?

"cate1" -> CatenateLayer[],
"Sres" -> FunctionLayer[Function[{#[[1]] + \[Beta]*#[[2]]*#[[3]]/N1}]],
"cate2" -> CatenateLayer[],
"Ires" -> 
 FunctionLayer[
  Function[{#[[1]] - \[Beta]*#[[2]]*#[[3]]/N1 + \[Gamma]*#[[3]]}]],
"cate3" -> CatenateLayer[],
"Rres" -> FunctionLayer[Function[{#[[1]] - \[Gamma]* #[[2]]}]],

I thought #[[1]],#[[2]], and #[[3]] should be used instead?

"cate1" -> CatenateLayer[],
"Sres" -> FunctionLayer[Function[{#[[1]] + \[Beta]*#[[2]]*#[[3]]/N1}]],
"cate2" -> CatenateLayer[],
"Ires" -> 
 FunctionLayer[
  Function[{#[[2]] - \[Beta]*#[[2]]*#[[3]]/N1 + \[Gamma]*#[[3]]}]],
"cate3" -> CatenateLayer[],
"Rres" -> FunctionLayer[Function[{#[[3]] - \[Gamma]* #[[2]]}]],

I found that the physical information neural network I made did not have a good prediction effect outside the training domain. How can I solve the problem?