Not able to save the values in rule input

Look on Appian example: https://docs.appian.com/suite/help/21.1/Editable_Grid_Component.html

hard to review entier code but your local!publication is Array of type type!{urn:com:appian:types:ISR}ISR_StudyDetail so you in below code you are trying to add to that array Today() +1 (save!value). 

If you want to append local!publication you need to add another object of type type!{urn:com:appian:types:ISR}ISR_StudyDetail

addRowlink: a!dynamicLink(
label: "Add Publication Details",
value: {
startDate: today() + 1
},
saveInto: {
a!save(local!publication, append(local!publication, save!value))
}
),

a!localVariables(
  local!publication: cast(
    typeof(
      'type!{urn:com:appian:types}studyDetail'()
    ),
    {}
  ),
  a!boxLayout(
    label: "Publicaton Plan",
    contents: {
      a!gridLayout(
        totalCount: count(local!publication),
        headerCells: {
          a!gridLayoutHeaderCell(label: "Publication Type"),
          a!gridLayoutHeaderCell(label: "Publication Name"),
          a!gridLayoutHeaderCell(label: "Year Estimate"),
          /* For the "Remove" column */
          a!gridLayoutHeaderCell(label: "")
        },
        columnConfigs: {
          a!gridLayoutColumnConfig(width: "DISTRIBUTE", weight: 3),
          a!gridLayoutColumnConfig(width: "DISTRIBUTE", weight: 3),
          a!gridLayoutColumnConfig(width: "DISTRIBUTE", weight: 3),
          a!gridLayoutColumnConfig(width: "ICON")
        },
        rows: a!forEach(
          items: local!publication,
          expression: a!gridRowLayout(
            contents: {
              a!radioButtonField(
                label: "Publication Type",
                value: fv!item.publicationType,
                saveInto: fv!item.publicationType,
                required: true(),
                choiceLabels: { "Conference", "Journal" },
                choiceValues: { "Conference", "Journal" },
                choiceLayout: "COMPACT"
              ),
              a!paragraphField(
                label: "Publication Name",
                /*instructions: rule!ISR_er_InstructionsText(*/
                /*data: fv!item.publicationName,*/
                /*maxLength: cons!ISR_PARAGRAPH_LENGTH*/
                /*),*/
                /*not sure why u need refresh*/
                /*refreshAfter: "KEYPRESS",*/
                value: fv!item.publicationName,
                saveInto: {
                  clean(fv!item.publicationName),
                  /*not sure on the study details type for now commenting*/
                  /*a!save(fv!item.publicationName,*/
                  /*ri!studyDetail.publicationName*/
                  /*),*/
                  /**/
                  
                },
                required: true(),
                /*validationGroup: cons!ISR_VALIDATION_GROUP_SUBMIT,*/
                /*validations: rule!ISR_er_ValidateTextLength(*/
                /*text: fv!item.publicationName,*/
                /*textLengthAllowed: cons!ISR_PARAGRAPH_LENGTH*/
                /*),*/
                
              ),
              a!dropdownField(
                label: "Year estimate",
                placeholder: "--- Select a Year ---",
                choiceLabels: cons!ISR_yearlistforpublication,
                choiceValues: cons!ISR_yearlistforpublication,
                value: fv!item.estimatedYear,
                saveInto: fv!item.estimatedYear,
                required: true()
              ),
              a!richTextDisplayField(
                value: a!richTextIcon(
                  icon: "close",
                  altText: "delete " & fv!index,
                  caption: "Remove " & fv!item.publicationType & " " & fv!item.publicationName,
                  link: a!dynamicLink(
                    value: fv!index,
                    saveInto: {
                      a!save(
                        local!publication,
                        remove(local!publication, save!value)
                      )
                    }
                  ),
                  linkStyle: "STANDALONE",
                  color: "NEGATIVE"
                )
              )
            },
            id: fv!index
          )
        ),
        addRowlink: a!dynamicLink(
          label: "Add Publication Details",
          saveInto: {
            a!save(
              local!publication,
              append(local!publication, save!value)
            )
          }
        ),
        rowHeader: 1
      )
    },
    iscollapsible: true,
    isInitiallyCollapsed: false(),
    marginBelow: "STANDARD"
  )
)

I tried to debug the code and added the comments as well. Check the above code whether it is working for your requirement